a) Đặt x - y = a. Ta có:
(x - y)2 + 4(x - y) - 12
= a2 + 4a - 12
= (a2 - 2a) + (6a - 12)
= a(a - 2) + 6(a - 2)
= (a + 6)(a - 2)
= (x + y + 6)(x + y - 2)
a)\(\left(x-y\right)^2+4\left(x-y\right)-12=\left(x-y\right)^2-2\left(x-y\right)+6\left(x-y\right)-12\)
....................................................\(=\left(x-y\right)\left(x-y-2\right)+6\left(x-y-2\right)\)
....................................................\(=\left(x-y+6\right)\left(x-y-2\right)\)
b)\(x^2+y^2+3x-3y-2xy-10\)
\(=\left(x-y\right)^2+3\left(x-y\right)-10\)
\(=\left(x-y\right)\left(x-y+3\right)-10\)
\(=z\left(z+3\right)-10\) với \(z=x-y\)
\(=z^2+3z-10\)
\(=z^2-2z+5z-10\)
\(=z\left(z-2\right)+5\left(z-2\right)\)
\(=\left(z+5\right)\left(z-2\right)\)
f)\(x^2-6x-16=x^2+2x-8x-16=x\left(x+2\right)-8\left(x+2\right)=\left(x-8\right)\left(x+2\right)\)
g)\(\left(x+2\right)\left(x+3\right)\left(x+4\right)\left(x+5\right)-24\)
\(=\left[\left(x+2\right)\left(x+5\right)\right]\left[\left(x+3\right)\left(x+4\right)\right]-24\)
\(=\left(x^2+7x+10\right)\left(x^2+7x+12\right)-24\)
\(=y\left(y+2\right)-24\) với \(y=x^2+7x+10\)
\(=y^2+2y-24\)
\(=y^2-4y+6y-24\)
\(=y\left(y-4\right)+6\left(y-4\right)\)
\(=\left(y+6\right)\left(y-4\right)\)
h)\(\left(x^2+6x+5\right)\left(x^2+10x+21\right)+15\)
\(=\left(x^2+x+5x+5\right)\left(x^2+3x+7x+21\right)+15\)
\(=\left[x\left(x+1\right)+5\left(x+1\right)\right]\left[x\left(x+3\right)+7\left(x+3\right)\right]+15\)
\(=\left(x+1\right)\left(x+5\right)\left(x+7\right)\left(x+3\right)+15\)
\(=\left[\left(x+1\right)\left(x+7\right)\right]\left[\left(x+3\right)\left(x+5\right)\right]+15\)
\(=\left(x^2+8x+7\right)\left(x^2+8x+15\right)+15\)
\(=y\left(y+8\right)+15\) với \(y=x^2+8x+7\)
\(=y^2+8y+15\)
\(=y^2+3y+5y+15\)
\(=y\left(y+3\right)+5\left(y+3\right)\)
\(=\left(y+3\right)\left(y+5\right)\)
b) Đặt x - y = b. Ta có:
x2 + y2 + 3x - 3y - 2xy - 10
= (x - y)2 + 3(x - y) - 10
= b2 + 3b - 10
= (b2 + 5b) - (2b + 10)
= b(b + 5) - 2(b + 5)
= (b - 2)(b + 5)
= (x - y - 2)(x - y + 5)
f) x2 - 6x - 16
= (x2 - 8x) + (2x - 16)
= x(x - 8) + 2(x - 8)
= (x + 2)(x - 8)
g) Đặt x2 + 7x + 10 = v. Ta có:
(x + 2)(x + 3)(x + 4)(x + 5) - 24
= [(x + 2)(x + 5)][(x + 3)(x + 4)] - 24
= [x2 + 7x + 10][x2 + 7x + 10 + 2] - 24
= v(v + 2) - 24
= v2 + 2v - 24
= (v2 - 4x) + (6x - 24)
= v(v - 4) + 6(v - 4)
= (v + 6)(v - 4)
= (x2 + 7x + 10 + 6)(x2 + 7x + 10 - 4)
= (x2 + 7x + 16)(x2 + 7x + 6)
