分解因式 （x^2+4x)(x^2+8x+12)+15

题目

分解因式 （x^2+4x)(x^2+8x+12)+15

答案

原式 = （x^2+4x)(x^2+8x+12)+15
= x^4 + 8x^3 + 12x^2 + 4x^3 + 32x^2 + 48x+15
= x^4 + 12x^3 + 44x^2 + 48x +15
= (x^4+x^3) + (11x^3+11x^2) + (33x^2+33x) + (15x+15)
= x^3(x+1) + 11x^2(x+1) + 33x(x+1) + 15(x+1)
= (x+1)(x^3+11x^2+33x+15)
= (x+1)[(x^3+5x^2) + (6x^2+30x) + (3x+15)]
= (x+1)[x^2(x+5) + 6x(x+5) + 3(x+5)]
= (x+1)(x+5)(x^2+6x+3)