1)
$x^4 +3x^3 +x^2 -12x -20$
$= x^4 - 4x^2 + 3x^3 - 12x + 5x^2 - 20$
$= x^2(x^2 - 4) + 3x(x^2 - 4) + 5(x^2 - 4)$
$= (x^2 + 3x + 5)(x^2 - 4)$
$= (x^2 + 3x + 5)(x - 2)(x + 2)$
2)
$x^3 - 10x - 12$
$= x^3 + 2x^2 - 2x^2 - 4x - 6x - 12$
$= x^2(x + 2) - 2x(x + 2) - 6(x + 2)$
$= (x^2 - 2x - 6)(x + 2)$
3)
Đề sai
4)
$x^8 + x + 1$
$= x^8 + x^7 + x^6 - x^7 - x^6 - x^5 + x^5 + x^4 + x^3 - x^4 - x^3 - x^2 + x^2 + x + 1$
$= x^6(x^2 + x + 1) - x^5(x^2 + x + 1) + x^3(x^2 + x + 1) - x^2(x^2 + x + 1) + x^2 + x + 1$
$= (x^6 - x^5 + x^3 - x^2 + 1)(x^2 + x + 1)$
5)
$x^8 + x^7 + 1$
$= x^8 + x^7 + x^6 - x^6 - x^5 - x^4 + x^5 + x^4 + x^3 - x^3 - x^2 - x + x^2 + x + 1$
$= x^6(x^2 + x + 1) - x^4(x^2 + x + 1) + x^3(x^2 + x + 1) - x(x^2 + x + 1) + (x^2 + x + 1)$
$= (x^6 - x^4 + x^3 - x + 1)(x^2 + x + 1)$
6)
$4x^4y^4 + 1$
$= 4x^4y^4 + 4x^2y^2 + 1 - 4x^2y^2$
$= (2x^2y^2)^2 + 2.(2x^2y^2) + 1 - 4x^2y^2$
$= (2x^2y^2 + 1)^2 - (2xy)^2$
$= (2x^2y^2 - 2xy + 1)(2x^2y^2 + 2xy + 1)$
7)
Đặt $t = x^2 + x$, ta có:
$(x^2 + x)^2 + 4x(x + 1) - 12$
$= t^2 + 4t - 12$
$= t^2 - 2t + 6t - 12$
$= (t - 2)t + (t - 2)6$
$= (t - 2)(t + 6)$
$= (x^2 + x - 2)(x^2 + x + 6)$
$= (x^2 -x + 2x - 2)(x^2 + x + 6)$
$= (x(x - 1) + 2(x - 1))(x^2 + x + 6)$
$= (x + 2)(x - 1)(x^2 + x + 6)$