A = $\Large{\frac{2xy(3x^2 - xy) + (6x^3y^3 - 12x^4y^2 + 3xy)}{3xy}}$
Áp dụng quy tắc: a+(b+c) = a+b+c
$2xy(3x^2−xy)+(6x^2y^2.3−12x^4y^2+3xy)$
A = $\Large{\frac{2xy(3x^2 − xy) + 6x^2y^2 − 12x^4y^2 + 3xy} {3xy}}$
Đặt xy làm thừa số chung, ta có:
A = $\Large{\frac{xy(6x^2 - 2xy + 6xy -12x^3y + 3)} {3xy}}$
A = $\Large{\frac{xy(6x^2 + 4xy - 12x^3y + 3)} {3xy}}$
Triệt tiêu x
A = $\Large{\frac{xy(6x^2 + 4xy - 12x^3y + 3)} {3xy}}$
6) $\Large{\frac{a-1}{a-2}+\frac{a-3}{a-2}}$
= $\Large{\frac{(a -1)+(a -3)}{a-2}}$
= $\Large{\frac{a -1+ a -3}{a -2}}$
= $\Large{\frac{2a - 4}{a -2}}$
= $\Large{\frac{2(a - 2)}{a -2}}$
`
= $2$
7) $(2x - 3)^2 + (x + 2)(1 - 4x)$ = 4x2 -12x + 9 + x - 4x2 +2 - 8x
= 4x2 - 4x2 -12x + x - 8x + 9 + 2 = - 19x +11
8) $\Large{\frac{6x}{x^2-9}+\frac{5x}{x-3}+\frac{x}{x+3}}$
A= $\Large{\frac{6x}{x^2-9}+\frac{5x}{x-3}+\frac{x}{x+3}}$
mà ${x^2-9}$ = $(x+3)(x-3)$
A = $\Large{\frac{6x}{(x+3)(x-3)}+\frac{5x(x+3)}{(x+3)(x-3)}+\frac{x(x-3)}{(x+3)(x-3)}}$
= $\Large{\frac{6x}{(x+3)(x-3)}+\frac{5x^2 +15x }{(x+3)(x-3)}+\frac{x^2 -3x }{(x+3)(x-3)}}$
= $\Large{\frac{6x + 5x^2 +15x + x^2 -3x }{(x+3)(x-3)}}$
= $\Large{\frac{5x^2 + x^2 + 15x + 6x -3x }{(x+3)(x-3)}}$
= $\Large{\frac{6x^2 + 18x}{(x+3)(x-3)}}$
= $\Large{\frac{6x(x + 3)}{(x+3)(x-3)}}$
= $\Large{\frac{6x}{x-3}}$
9) $5xy(2x^2 - xy) + (10x^3y^3 + 15x^4y^2 - 5xy): (5xy)$
A = $5xy(2x^2 - xy) + (10x^3y^3 + 15x^4y^2 - 5xy): (5xy)$
$(10x^3y^3 + 15x^4y^2 - 5xy): (5xy)$
= $5xy(x^2y^2 + 3x^3y -1) : (5xy)$
Triệt tiêu 5xy
= $x^2y^2 + 3x^3y -1$
A = $5xy(2x^2 - xy) + x^2y^2 + 3x^3y -1$
A = $10x^3y - 5x^2y^2 + x^2y^2 + 3x^3y -1$
A = $13x^3y - 4x^2y^2 - 1$
10) $\Large{\frac{3x-1}{x+2}+\frac{2x+11}{x+2}}$
A = $\Large{\frac{3x-1}{x+2}+\frac{2x+11}{x+2}}$
A = $\Large{\frac{3x-1 + 2x+11}{x+2}}$
A = $\Large{\frac{5x +10}{x+2}}$
A = $\Large{\frac{5(x +2)}{x+2}}$
Triệt tiêu x + 2
A = 5
11) $(2x - 1)^2+ (4x + 2)(1 - x)$
A = $(4x^2 - 4x +1) + (4x - 4x^2 + 2 - 2x)$
A = $4x^2 - 4x +1 + 4x - 4x^2 + 2 - 2x$
A = $ 4x^2 - 4x^2 - 4x + 4x - 2x +1 + 2$ A = $-2x +3 $
12) $3x^2y(x^2 - 4y) + (30x^5y^3 - 25x^3y^4 - 5xy^2) : 5xy^2$
A = $3x^2y(x^2 - 4y) + (30x^5y^3 - 25x^3y^4 - 5xy^2) : 5xy^2$
A = $(3x^4y - 12x^2y^2) + 5xy^2(6x^4y - 5x^2y^2 - 1) : 5xy^2$
A = $(3x^4y - 12x^2y^2) + (6x^4y - 5x^2y^2 - 1) $
A = $3x^4y - 12x^2y^2 + 6x^4y - 5x^2y^2 - 1 $
A = $3x^4y + 6x^4y - 12x^2y^2 - 5x^2y^2 - 1 $
A = $9x^4y - 17x^2y^2 - 1 $
13) $\Large{\frac{11x}{2x-3}+ \frac{x-18}{2x-3}}$
A = $\Large{\frac{11x}{2x-3}+ \frac{x-18}{2x-3}}$
A = $\Large{\frac{11x + x - 18}{2x - 3}}$
A = $\Large{\frac{12x - 18}{2x - 3}}$
A = $\Large{\frac{6(x - 3)}{2x - 3}}$
Triệt tiêu 2x - 3
A = 6
14) $(3x + 2)(3x - 2) + 9x(1 - x)$
1. Use Difference of Squares: $a2−b2=(a+b)(a−b)$.
$(3x)^2 − 2^2 + 9x(1 − x)$
2. Expand by distributing terms.
$(3x)^2 – 2^2 + 9x − 9x^2$
3. Use Multiplication Distributive Property: $(xy)a=xaya$.
$3^2x^2 − 2^2 + 9x − 9x^2$
4. Simplify $3^2$ to 9.
$9x^2 – 2^2 + 9x − 9x^2$
5. Simplify $2^2$ to 4.
$9x^2 – 4 + 9x − 9x^2$
6. Collect like terms.
$(9x^2 − 9x^2) – 4 + 9x$
7. Simplify.
$−4 + 9x$
8. Regroup terms.
$9x − 4$
15) $\Large{\frac{x}{x+2}-\frac{3}{x-2}-\frac{x^2+9}{x^2-4}}$
$x^2 - 4 = (x - 2)(x + 2)$
A = $\Large{\frac{x}{x+2}-\frac{3}{x-2}-\frac{x^2+9}{x^2-4}}$
A = $\Large{\frac{x(x-2) - 3(x+2) - (x^2+9)}{(x - 2)(x + 2)}}$
A = $\Large{\frac{x^2 - 2x - 3x - 6 - x^2 - 9)}{(x - 2)(x + 2)}}$
A = $\Large{\frac{x^2 - x^2 - 2x - 3x - 6 - 9}{(x - 2)(x + 2)}}$
A = $\Large{\frac{ - 5x - 15}{(x - 2)(x + 2)}}$
A = $\Large{\frac{ - 5(x + 3)}{(x - 2)(x + 2)}}$
16) $4x^2y(2xy^2 - xy)+ (25x^4y^3 - 10x^4y^4 + 5xy): (5xy)$
A = $4x^2y(2xy^2 - xy)+ (25x^4y^3 - 10x^4y^4 + 5xy): (5xy)$
A = $(8x^3y^3 - 4x^3y^3)+ (5xy(5x^3y^2 - 2x^3y^3 + 1)): (5xy)$
A = $8x^3y^3 - 4x^3y^3+ 5x^3y^2 - 2x^3y^3 + 1$
A = $8x^3y^3 - 4x^3y^3 - 2x^3y^3 + 5x^3y^2 + 1$
A = $8x^3y^3 - 4x^3y^3 - 2x^3y^3 + 5x^3y^2 + 1$
A = $2x^3y^3 + 5x^3y^2 + 1$
17) $\Large{\frac{2x-4}{3x+2}+\frac{3x+1}{3x+2}}$
A = $\Large{\frac{2x-4}{3x+2}+\frac{3x+1}{3x+2}}$
A = $\Large{\frac{2x - 4 + 3x+1}{3x+2}}$
A = $\Large{\frac{5x - 3}{3x+2}}$
18) $(x - 6)^2 + (x + 7)(4 - x)$
A = $(x - 6)^2 + (x + 7)(4 - x)$
A = $(x^2 - 12x + 36) + (4x - x^2 +28 - 7x)$
A = $x^2 - 12x + 36 + 4x - x^2 +28 - 7x$
A = $x^2 - x^2- 12x + 4x - 7x + 36 +28 $
A = $- 15x + 60 $
A = $- 15(x - 4) $
19) $\Large{\frac{1-3x}{2x}+\frac{3x-2}{2x-1}-\frac{3x-2}{4x^2-2x}}$
A = $\Large{\frac{1-3x}{2x}+\frac{3x-2}{2x-1}-\frac{3x-2}{4x^2-2x}}$
$4x^2-2x = 2x(2x -1)$
A = $\Large{\frac{(1-3x)(2x -1)}{2x(2x -1)}+\frac{2x(3x-2)}{2x(2x-1)}-\frac{3x-2}{2x(2x -1)}}$
A = $\Large{\frac{(1-3x)(2x -1) + 2x(3x-2) - (3x-2)}{2x(2x -1)}}$
A = $\Large{\frac{(2x -1 - 6x^2 + 3x) + (6x^2 - 4x) - (3x-2)}{2x(2x -1)}}$
A = $\Large{\frac{2x -1 - 6x^2 + 3x + 6x^2 - 4x - 3x + 2}{2x(2x -1)}}$
A = $\Large{\frac{ - 6x^2 + 6x^2+ 3x + 2x - 4x - 3x + 2 -1}{2x(2x -1)}}$
A = $\Large{\frac{ (- 6x^2 + 6x^2)+ (3x + 2x - 4x - 3x) + (2 -1)}{2x(2x -1)}}$
A = $\Large{\frac{- 2x + 1 }{2x(2x -1)}}$
A = $\Large{\frac{- (2x - 1) }{2x(2x -1)}}$
A = $\Large{\frac{- 1 }{2x }}$
20) $2xy(2x - xy^2) + (3x^4y^4 - 9x^4y^2 + 6x^2y): (3x^2y)$
A = $2xy(2x - xy^2) + (3x^4y^4 - 9x^4y^2 + 6x^2y): (3x^2y)$
A = $(4x^2y - 2x^2y^3) + (3x^2y(x^2y^3 - 3x^2y + 2): (3x^2y)$
Triệt tiêu $3x^2y$
A = $(4x^2y - 2x^2y^3) + (x^2y^3 - 3x^2y + 2)$
A = $4x^2y - 2x^2y^3 + x^2y^3 - 3x^2y + 2$ A = $ 4x^2y - 3x^2y + x^2y^3 - 2x^2y^3 + 2$
A = $- x^2y^3 + x^2y + 2$
21) $\Large{\frac{x+2}{x-1}+\frac{3x+1}{x-1}}$
A = $\Large{\frac{x+2}{x-1}+\frac{3x+1}{x-1}}$
A = $\Large{\frac{x + 2 + 3x + 1}{x-1}}$
A = $\Large{\frac{5x + 3}{x-1}}$
A = $27x -3$
22) $5x(5 - x) + (x + 1)(5x - 3)$
A = $5x(5 - x) + (x + 1)(5x - 3)$
A = $(25x - 5x^2) + (5x^2 - 3x + 5x -3 )$
A = $25x - 5x^2 + 5x^2 - 3x + 5x -3 $
A = $5x^2 - 5x^2 + 25x + 5x - 3x -3 $
A = $27x -3 $
23) $\Large{\frac{x}{x-2}-\frac{x}{x+2}-\frac{8}{x^2-4}}$
$x^2 - 4 = (x -2 ) (x + 2)$ A = $\Large{\frac{x}{x-2}-\frac{x}{x+2}-\frac{8}{ (x -2 ) (x + 2)}}$
A = $\Large{\frac{x(x+2) - x(x-2) - 8}{ (x -2 ) (x + 2)}}$
A = $\Large{\frac{x^2 + 2x - x^2 + 2x - 8}{ (x -2 ) (x + 2)}}$
A = $\Large{\frac{x^2 - x^2 + 2x + 2x - 8}{ (x -2 ) (x + 2)}}$
A = $\Large{\frac{ 4x - 8}{ (x -2 ) (x + 2)}}$
A = $\Large{\frac{ 4(x - 2)}{ (x -2 ) (x + 2)}}$
Triệt tiêu x-2
A = $\Large{\frac{ 4}{ x + 2}}$
24) $3y^2(2x^2 + xy) + (9x^2y^5 + 6x^3y^4 - 12xy^2): 3xy^2$ A
= $(6x^2y^2 + 3xy^3) + (9x^2y^5 + 6x^3y^4 - 12xy^2): 3xy^2$
$(9x^2y^5 + 6x^3y^4 - 12xy^2): 3xy^2$
= $3xy^2 (3xy^3 + 2x^2y^2 - 4): 3xy^2$
Triệt tiêu $3xy^2$
= $3xy^4 + 2x^2y^3 - 4$
A = $3y^2(2x^2 + xy) + 3xy^4 + 2x^2y^3 - 4$
A = $6x^2y^2 + 3xy^3 + 3xy^4 + 2x^2y^3 - 4$
A = $6x^2y^2 + 3xy^3 + 3xy^4 + 2x^2y^3 - 4$
25) $\Large{\frac{2x+3}{x+1}+\frac{3x-1}{x+1}}$
A = $\Large{\frac{(2x+3)+(3x-1)}{x+1}}$
A = $\Large{\frac{2x+3+3x-1}{x+1}}$
A = $\Large{\frac{5x+2}{x+1}}$
26) $(2x - 5)(2x + 5) - (1 - 2x)^2 - 4x$
A = $(2x - 5)(2x + 5) - (1 - 2x)^2 - 4x$
A = $((2x)^2 - 5^2) - (1 - 2x)^2 - 4x$
A = $(4x^2 - 25) - (1^2 - 2.1.2x + (2x)^2 ) - 4x$
A = $4x^2 - 25 - 1^2 + 2.1.2x - (2x)^2 - 4x$
A = $4x^2 - 25 - 1^2 + 2.1.2x - (2x)^2 - 4x$
A = $4x^2 - 25 - 1 + 4x - 4x^2 - 4x$ A = $(4x^2 - 4x^2) + (4x - 4x) - 25 - 1$
A = $- 25 - 1$
A = $ - 26$
27) $\Large{\frac{2}{x+1}+\frac{4}{x-1}-\frac{5x+1}{x^2-1}}$
$x^2-1$ = $x^2-1^2$ = $(x-1)(x+1)$
A = $\Large{\frac{2}{x+1}+\frac{4}{x-1}-\frac{5x+1}{x^2-1}}$
A = $\Large{\frac{2(x-1) + 4(x+1) - (5x+1)}{(x-1)(x+1)}}$
A = $\Large{\frac{2x - 2 + 4x + 4 - 5x -1}{(x-1)(x+1)}}$ A
= $\Large{\frac{2x + 4x - 5x - 2 + 4 -1}{(x-1)(x+1)}}$
A = $\Large{\frac{x +1}{(x-1)(x+1)}}$
A = $\Large{\frac{x +1}{(x-1)(x+1)}}$
Triệt tiêu x + 1
A = $\Large{\frac{1}{(x - 1)}}$
28) $5xy(4x^2 - xy) + (4x^5y^2 - 8x^4y^3 + 16x^2y): (4x^2y)$
$(4x^5y^2 - 8x^4y^3 + 16x^2y): (4x^2y)
= 4x^2y(x^3y - 2x^2y^2 + 4): (4x^2y)$
Triệt tiêu $(4x^2y)$ = $x^3y - 2x^2y^2 + 4$
A = $5xy(4x^2 - xy) + (4x^5y^2 - 8x^4y^3 + 16x^2y): (4x^2y)$
A = $5xy(4x^2 - xy) + x^3y - 2x^2y^2 + 4$
A = $(20x^3y - 5x^2y^2) + x^3y - 2x^2y^2 + 4$
A = $20x^3y - 5x^2y^2 + x^3y - 2x^2y^2 + 4$
A = $(20x^3y + x^3y)- (5x^2y^2 + 2x^2y^2) + 4$
A = $21x^3y - 7x^2y^2 + 4$
29) $\Large{\frac{2x-1}{3x-2}+\frac{x+3}{3x-2}}$
A = $\Large{\frac{2x-1}{3x-2}+\frac{x+3}{3x-2}}$
A = $\Large{\frac{(2x-1)+(x+3)}{3x-2}}$
A = $\Large{\frac{2x-1+x+3}{3x-2}}$
A = $\Large{\frac{3x+2}{3x-2}}$
30) $(x + 5)^2 - (x + 3). (x - 2)$
A = $(x + 5)^2 - (x + 3). (x - 2)$
A = $(x^2 + 2.5.x + 5^2) - (x^2 - 2x + 3x - 6)$
A = $x^2 + 2.5.x + 5^2 - x^2 + 2x - 3x + 6$
A = $(x^2 - x^2) + (2.5.x + 2x - 3x) + (5^2 + 6)$
A = $ 9x + 31$
31) $\Large{\frac{1}{x+2}+\frac{2}{x-2}-\frac{2x}{x^2-4}}$
$x^2 - 4$ = $x^2 - 2^2$ = $(x - 2)(x + 2)$
A = $\Large{\frac{1}{x+2}+\frac{2}{x-2}-\frac{2x}{x^2-4}}$
A = $\Large{\frac{1(x - 2) + 2(x + 2) - 2x}{(x - 2)(x + 2)}}$
A = $\Large{\frac{x - 2 + 2x + 4 -2x}{(x - 2)(x + 2)}}$
A = $\Large{\frac{x + 2}{(x - 2)(x + 2)}}$
Triệt tiêu x +2
A = $\Large{\frac{1}{(x - 2)}}$
32) $\Large{\frac{x^2 - 1}{x^2 + 4x}.\frac{2x}{x - 1}}$
A = $\Large{\frac{x^2 - 1}{x^2 + 4x}.\frac{2x}{x - 1}}$
A = $\Large{\frac{(x^2 - 1)2x}{(x^2 + 4x)(x - 1)}}$
A = $\Large{\frac{2x^3 - 2x}{x^3 - x^2 + 4x^2 - 4x}}$
A = $\Large{\frac{2x^3 - 2x}{x^3 + 3x^2 - 4x}}$
A = $\Large{\frac{2x(x^2 - 1)}{x(x^2 + 3x - 4)}}$
Triệt tiêu x
A = $\Large{\frac{2(x^2 - 1)}{(x^2 + 3x - 4)}}$
33) $\Large{\frac{x+y}{xy}+\frac{x-2}{xy}+\frac{x-14}{x^2-4}}$
$x^2-4 = x^2 - 2^2$ = $(x+2)(x-2)$
A = $\Large{\frac{x+y}{xy}+\frac{x-2}{xy}+\frac{x-14}{x^2-4}}$
A = $\Large{\frac{x+y}{xy}+\frac{x-2}{xy}+\frac{x-14}{x^2 - 2^2}}$
A = $\Large{\frac{x+y}{xy}+\frac{x-2}{xy}+\frac{x-14}{(x+2)(x-2)}}$
MTC: $xy(x+2)(x-2)$
A = $\Large{\frac{(x+y)(x+2)(x-2)+(x-2)(x+2)(x-2)+(x-14)xy}{xy(x+2)(x-2)}}$
$x^ax^b = x^ab$
A = $\Large{\frac{(x+y)(x+2)(x-2)+((x-2)(x-2))(x+2)+(x-14)xy}{xy(x+2)(x-2)}}$
A = $\Large{\frac{(x+y)(x+2)(x-2)+(x-2)^2(x+2)+(x-14)xy}{xy(x+2)(x-2)}}$
A = $\Large{\frac{x^3−2x^2+2x^2−4x+yx^2−2yx+2yx−4y+x3+2x^2−4x^2−8x+4x+8+x2y−14xy}{xy(x+2)(x-2)}}$
A = $\Large{\frac{(x^3+x^3)+(−2x^2+2x^2+2x^2−4x^2)+(−4x−8x+4x)+(x^2y+x^2y)+(−2xy+2xy−14xy)−4y+8}{xy(x+2)(x-2)}}$ $(x^3+x^3)+(−2x^2+2x^2+2x^2−4x^2)+(−4x−8x+4x)+(x^2y+x^2y)+(−2xy+2xy−14xy)−4y+8$
= $2x^3−2x^2−8x+2x^2y−14xy−4y+8$
A = $\Large{\frac{2x^3−2x^2−8x+2x^2y−14xy−4y+8}{xy(x+2)(x-2)}}$
A = $\Large{\frac{2(x^3−x^2−4x+x^2y−7xy−2y+4)}{xy(x+2)(x-2)}}$
34) $\Large{\frac{x+1}{x-2}-\frac{x-2}{x+2}+\frac{x-14}{x^2-4}}$
$x^2-4 = x^2-2^2=(x+2)(x-2)$
A = $\Large{\frac{x+1}{x-2}-\frac{x-2}{x+2}+\frac{x-14}{x^2-4}}$
A = $\Large{\frac{x+1}{x-2}-\frac{x-2}{x+2}+\frac{x-14}{(x+2)(x-2)}}$
A = $\Large{\frac{(x+1)(x+2)−(x−2)(x−2)+x−14}{(x+2)(x-2)}}$
$x^ax^b=x^{a+b}.$
A = $\Large{\frac{(x+1)(x+2)−(x−2)^2+x−14}{(x+2)(x-2)}}$
A = $\Large{\frac{x^2+2x+x+2−x^2+2x.2−2^2+x−14}{(x+2)(x-2)}}$
$2^2 = 4$ A = $\Large{\frac{x^2+2x+x+2−x^2+2x.2−4+x−14}{(x+2)(x-2)}}$
$2x.2=4x$
A = $\Large{\frac{x^2+2x+x+2−x^2+4x−4+x−14}{(x+2)(x-2)}}$
A = $\Large{\frac{(x^2−x^2)+(2x+x+4x+x)+(2−4−14)}{(x+2)(x-2)}}$
$(x^2−x^2)+(2x+x+4x+x)+(2−4−14) = 8x−16$
A = $\Large{\frac{8x−16}{(x+2)(x-2)}}$
A = $\Large{\frac{8(x−2)}{(x+2)(x-2)}}$
Triệt tiêu x-2
A = $\Large{\frac{8 }{(x+2) }}$
35) $\Large{\frac{1}{x+5}-\frac{1}{x-5}+\frac{2x}{x^2-25}}$$a^2-b^2; a=x, b=5$
$x^2-25 = x^2-5^2=(x+5)(x-5)$ A = $\Large{\frac{1}{x+5}-\frac{1}{x-5}+\frac{2x}{x^2-25}}$
A = $\Large{\frac{1}{x+5}-\frac{1}{x-5}+\frac{2x}{(x+5)(x-5)}}$
A = $\Large{\frac{x−5−(x+5)+2x}{(x+5)(x-5)}}$
A = $\Large{\frac{x−5−x-5+2x}{(x+5)(x-5)}}$
A = $\Large{\frac{(x−x+2x)+(−5−5)}{(x+5)(x-5)}}$
A = $\Large{\frac{(2x-10)}{(x+5)(x-5)}}$
A = $\Large{\frac{(2(x-5))}{(x+5)(x-5)}}$
Triệt tiêu x-5
A = $\Large{\frac{2)}{x+5}}$
36) $6xy(2x^2 -3y)+(4x^2y^3 – 2x^4y^3 – 2x^4y^2 + 3xy):(xy)$ ?
37) $\Large{\frac{a^2+2b}{a+3}+\frac{3a-2b}{a+3}}$
A = $\Large{\frac{a^2+2b}{a+3}+\frac{3a-2b}{a+3}}$
A = $\Large{\frac{a^2+2b+3a−2b}{a+3}}$
A = $\Large{\frac{a(a+3)}{a+3}}$ A = a
38) $(3x +1)^2 –(3x -2)(3x + 4)$$(a+b)^2=a^2+2ab+b^2$
A = $(3x +1)^2 –(3x -2)(3x + 4)$
A = $(3x)^2+2×3x+1−(3x−2)(3x+4)$
A = $(3x)^2+2×3x+1−(9x^2+12x−6x−8)$
A = $(3x)^2+2×3x+1−9x^2-12x+6x+8)$
$(xy)^a=x^ay^a.$
A = $32x^2+2×3x+1−9x^2−12x+6x+8$
$3^2=9$
A = $9x^2+2×3x+1−9x^2−12x+6x+8$
$2.3x = 6x$
A = $9x^2+6x+1−9x^2−12x+6x+8$
A = $(9x^2−9x^2)+(6x−12x+6x)+(1+8)$
A = 9
39) $\Large{\frac{x+1}{x-2}+\frac{4}{x+2}+\frac{2-7x}{x^2-4}}$
$a^2 - b^2 =(a+b)(a-b)$
a = x; b = 2
$x^2 − 2^2 = (x + 2)(x - 2)$$\Large{\frac{x+1}{x−2}+\frac{4x}{x+2}+\frac{2−7x}{x^2−2^2}}$
$\Large{\frac{x+1}{x−2}+\frac{4x}{x+2}+\frac{2−7x}{(x + 2)(x - 2)}}$
$\Large{\frac{(x+1)(x+2)+4x(x−2)+2−7x}{(x + 2)(x - 2)}}$
$\Large{\frac{(x+1)(x+2)+4x(x−2)+2−7x}{(x + 2)(x - 2)}}$
$\Large{\frac{x^2+2x+x+2+4x^2−8x+2−7x}{(x + 2)(x - 2)}}$
$\Large{\frac{(x^2+4x^2)+(2x+x−8x−7x)+(2+2)}{(x + 2)(x - 2)}}$
$\Large{\frac{(5x^2−12x+4}{(x + 2)(x - 2)}}$
$\Large{\frac{5x^2−2x−10x+4}{(x + 2)(x - 2)}}$
$\Large{\frac{5x^2−2x−10x+4}{(x + 2)(x - 2)}}$
$\Large{\frac{x(5x−2)−2(5x−2)}{(x + 2)(x - 2)}}$
$\Large{\frac{(5x−2)(x−2)}{(x + 2)(x - 2)}}$
$\Large{\frac{5x−2}{(x + 2)(x - 2)}}$
40) $5a^2b(2a - ab^2) + (9a^5b^5 – 12a^5b^3 + 15a^2b^2): (3a^2b^2)$
41) $\Large{\frac{5x-2}{4x+12}+\frac{x+9}{4x+12}}$
42) $(2x + l)^2 - (2x + 3)(x - 1)$
43) $\Large{\frac{1}{x+5}+\frac{6}{x-5}-\frac{2x}{x^2-25}}$
44) $3x (2x^2y+ 3xy) + (6x^4y^2 - 8x^3y^2 + 4xy) : (2xy)$
45) $\Large{\frac{x^2-2x}{x+3}:\frac{x^2-4}{x^2+3x}}$
46) $(x - 3)(x + 3) + (x - 5)^2 - 2x(x - 5)$
47) $\Large{\frac{5x+6}{x^2-4}+\frac{2}{x+2}-\frac{4}{x-2}}$
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