1. Développer les expressions suivantes :

- a. \( 2(3x - 1)(2 - x) \)

\[
2(3x - 1)(2 - x) = (6x - 2)(2 - x)
\]
\[
= 12x - 6x^2 - 4 + 2x
\]
\[
= -6x^2 + 14x - 4
\]

- b. \( (2x + 3)^2 \)

\[
(2x + 3)^2 = 4x^2 + 12x + 9
\]

- c. \( (3x - 2)(3x + 2) \)

\[
(3x - 2)(3x + 2) = (3x)^2 - 2^2
\]
\[
= 9x^2 - 4
\]

- d. \( (5x - 6)^2 \)

\[
(5x - 6)^2 = (5x)^2 - 2 \times 5x \times 6 + 6^2
\]
\[
= 25x^2 - 60x + 36
\]

2. Factoriser les expressions suivantes :

- a. \( (x + 1)(1 - x) - (x + 1)(2x + 1) \)

\[
(x + 1)(1 - x) - (x + 1)(2x + 1) = (x + 1)[(1 - x) - (2x + 1)]
\]
\[
= (x + 1)(1 - x - 2x - 1)
\]
\[
= (x + 1)(-3x)
\]
\[
= -3x(x + 1)
\]

- b. \( 3(2x - 2) + (x + 1)(1 - x) \)

\[
3(2x - 2) + (x + 1)(1 - x) = 3[2(x - 1)] + (x + 1)[-(x - 1)]
\]
\[
= 6(x - 1) - (x + 1)(x - 1)
\]
\[
= (x - 1)[6 - (x + 1)]
\]
\[
= (x - 1)(6 - x - 1)
\]
\[
= (x - 1)(5 - x)
\]

- c. \( 2x(x + 1) + (x + 1)(x^2 + 1) \)

\[
2x(x + 1) + (x + 1)(x^2 + 1) = (x + 1)[2x + (x^2 + 1)]
\]
\[
= (x + 1)(x^2 + 2x + 1)
\]
\[
= (x + 1)(x + 1)^2
\]

- d. \( 12x^2 - 6x + (2x - 1)(5 - 2x) \)

\[
12x^2 - 6x + (2x - 1)(5 - 2x) = 6x(2x - 1) + (2x - 1)(5 - 2x)
\]
\[
= (2x - 1)[6x + (5 - 2x)]
\]
\[
= (2x - 1)(4x + 5)
\]

Effectuer les factorisations suivantes :

a. \( (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) \)}

On remarque l'égalité : \( 2 - 2x = -2(x - 1) \)

\[
(3x + 1)(2 - 2x) - (5 - 4x)(x - 1) = (3x + 1)[-2(x - 1)] - (5 - 4x)(x - 1)
\]
\[
= (x - 1)[-2(3x + 1) - (5 - 4x)]
\]
\[
= (x - 1)(-6x - 2 - 5 + 4x) = (x - 1)(-2x - 7)
\]

b. \( (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) \)}

On a l'égalité : \( -6x - 9 = -3(2x + 3) \)

\[
(2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) = (2 - 3x)(3 + 2x) + (3x + 2)[-3(2x + 3)]
\]
\[
= (2x + 3)[(2 - 3x) - 3(3x + 2)]
\]
\[
= (2x + 3)(2 - 3x - 9x - 6)
\]
\[
= (2x + 3)(-12x - 4)
\]
\[
= -4(2x + 3)(3x + 1)
\]

c. \( (6x + 2)(2x + 3) + (9x + 3)^2 \)}

On a : \( 6x + 2 = 2(3x + 1) \) ; \( 9x + 3 = 3(3x + 1) \)

\[
(6x + 2)(2x + 3) + (9x + 3)^2 = [2(3x + 1)(2x + 3)] + [3(3x + 1)]^2
\]
\[
= (3x + 1)[2(2x + 3) + 9(3x + 1)]
\]
\[
= (3x + 1)(4x + 6 + 27x + 9)
\]
\[
= (3x + 1)(31x + 15)
\]

d. \( (3x + 3)^2 - (x + 2)(5x + 4) \)}

\[
(3x + 3)^2 - (x + 2)(5x + 4) = (9x^2 + 18x + 9) - (5x^2 + 4x + 10x + 8)
\]
\[
= 9x^2 + 18x + 9 - 5x^2 - 14x - 8
\]
\[
= 4x^2 + 4x + 1 = (2x + 1)^2
\]

Effectuer les factorisations suivantes : 

a. \( (3x + 1)(2 - 2x) - (5 - 4x)(x - 1) \)}

On remarque l'égalité : \( 2 - 2x = -2(x - 1) \)

\[
(3x + 1)(2 - 2x) - (5 - 4x)(x - 1) = (3x + 1)[-2(x - 1)] - (5 - 4x)(x - 1)
\]
\[
= (x - 1)[-2(3x + 1) - (5 - 4x)]
\]
\[
= (x - 1)(-6x - 2 - 5 + 4x) = (x - 1)(-2x - 7)
\]

b. \( (2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) \)}

On a l'égalité : \( -6x - 9 = -3(2x + 3) \)

\[
(2 - 3x)(3 + 2x) + (3x + 2)(-6x - 9) = (2 - 3x)(3 + 2x) + (3x + 2)[-3(2x + 3)]
\]
\[
= (2x + 3)[(2 - 3x) - 3(3x + 2)]
\]
\[
= (2x + 3)(2 - 3x - 9x - 6)
\]
\[
= (2x + 3)(-12x - 4)
\]
\[
= -4(2x + 3)(3x + 1)
\]

c. \( (6x + 2)(2x + 3) + (9x + 3)^2 \)}

On a : \( 6x + 2 = 2(3x + 1) \) ; \( 9x + 3 = 3(3x + 1) \)

\[
(6x + 2)(2x + 3) + (9x + 3)^2 = [2(3x + 1)](2x + 3) + [3(3x + 1)]^2
\]
\[
= (3x + 1)[2(2x + 3) + 9(3x + 1)]
\]
\[
= (3x + 1)(4x + 6 + 27x + 9)
\]
\[
= (3x + 1)(31x + 15)
\]

d. \( (3x + 3)^2 - (x + 2)(5x + 4) \)}

\[
(3x + 3)^2 - (x + 2)(5x + 4) = (9x^2 + 18x + 9) - (5x^2 + 4x + 10x + 8)
\]
\[
= 9x^2 + 18x + 9 - 5x^2 - 14x - 8
\]
\[
= 4x^2 + 4x + 1 = (2x + 1)^2
\]

Factoriser les expressions suivantes : 

- a. \( (x-1)(2x+1) - (2x-2)(5-2x) \)

\[
(x-1)(2x+1) - (2x-2)(5-2x) = (x-1)(2x+1) - 2(x-1)(5-2x)
\]
\[
= (x-1)[(2x+1) - 2(5-2x)]
\]
\[
= (x-1)(2x+1 - 10 + 4x)
\]
\[
= (x-1)(6x - 9)
\]
\[
= 3(x-1)(2x - 3)
\]

- b. \( (2+x)(3-x) + (5-2x)(3-x) \)

\[
(2+x)(3-x) + (5-2x)(3-x) = [(2+x) + (5-2x)](3-x)
\]
\[
= (2 + x + 5 - 2x)(3 - x)
\]
\[
= (7 - x)(3 - x)
\]

- c. \( 3(4+2x) - (3+x)(10+5x) \)

\[
3(4+2x) - (3+x)(10+5x) = 3[2(2+x)] - (3+x)[5(2+x)]
\]
\[
= 6(2+x) - 5(3+x)(2+x)
\]
\[
= [6 - 5(3+x)](2+x)
\]
\[
= (6 - 15 - 5x)(2+x)
\]
\[
= 5(-9 - 5x)(2+x)
\]
\[
= -(5x + 9)(x + 2)
\]

- d. \( (2-x)(3x-4) + \left(2-\frac{3}{2}x\right)(2x+3) \)

\[
(2-x)(3x-4) + \left(2-\frac{3}{2}x\right)(2x+3) = (2-x)(3x-4) + \left[-\frac{1}{2}(-4+3x)\right](2x+3)
\]
\[
= (2-x)(3x-4) - \frac{1}{2}(-4+3x)(2x+3)
\]
\[
= \left[(2-x) - \frac{1}{2}(2x+3)\right](3x-4)
\]
\[
= \left(2 - x - x - \frac{3}{2}\right)(3x-4)
\]
\[
= \left(\frac{1}{2} - 2x\right)(3x-4)
\]

On pouvait trouver également \(\left(2-\frac{3}{2}x\right)(4x-1)\) qui est une expression égale.

- e. \( (2x+1)^2 - 4(2-3x)^2 \)

\[
(2x+1)^2 - 4(2-3x)^2 = (2x+1)^2 - [2(2-3x)]^2
\]
\[
= [(2x+1) + 2(2-3x)][(2x+1) - 2(2-3x)]
\]
\[
= (2x+1 + 4 - 6x)(2x+1 - 4 + 6x)
\]
\[
= (5 - 4x)(8x - 3)
\]

- f. \( 18x^2 - 24x + 8 + (3x-2)(2-x) \)

\[
18x^2 - 24x + 8 + (3x-2)(2-x) = 2(9x^2 - 12x + 4) + (3x-2)(2-x)
\]
\[
= 2(3x-2)^2 + (3x-2)(2-x)
\]
\[
= (3x-2)\left[2(3x-2) + (2-x)\right]
\]
\[
= (3x-2)(6x - 4 + 2 - x)
\]
\[
= (3x-2)(5x - 2)
\]