把下列多项式因式分解

题目

把下列多项式因式分解
(x+1)(x+2)(x+3)(x+4)-24 3a²b²-17abxy+10x²y² (x²-2x)²-7(x²-2x)+12 a(a+m)-b(b+m) (x²+2x+2)(x²+2x-3)+4

答案

(x+1)(x+2)(x+3)(x+4)-24
=[(x+1)(x+4)][(x+2)(x+3)]-24
=(x^2+5x+4)(x^2+5x+6)-24
=(x^2+5x+4)^2+2(x^2+5x+4)-24
=(x^2+5x+4+6)(x^2+5x+4-4)
=(x^2+5x+10)(x^2+5x)
=x(x+5)(x^2+5x+10)
3a²b²-17abxy+10x²y²
=(3ab-2xy)(ab-5xy)
(x²-2x)²-7(x²-2x)+12
=(x^2-2x-3)(x^2-2x-4)
=(x-3)(x+1)(x^2-2x-4)
a(a+m)-b(b+m)
=a^2+am-b^2-bm
=a^2-b^2+am-bm
=(a^2-b^2)+m(a-b)
=(a+b)(a-b)+m(a-b)
=(a+b)(a-b+m)
(x²+2x+2)(x²+2x-3)+4
=(x^2+2x)^2-(x^2+2x)-2
=(x^2+2x-2)(x^2+2x+1)
=(x^2+2x-2)(x+1)^2