Question

谁能帮我解一道因式分解的题目

（x+1)(x+3)(x+5)(x+7)+15

Answer 1

解：因为：
(x+3)(x+5)=x^2+8x+15
(x+1)(x+7)=x^2+8x+7
设(x+3)(x+5)＝y
则：(x+1)(x+7)＝y-8

原方程化为 y(y-8)+15=0
y^2-8y+15=0
(y-3)(y-5)=0
y=3或y=5

所以：(x+3)(x+5)＝3或(x+3)(x+5)＝5
当：(x+3)(x+5)＝3时
x^2+8x+15=3
x^2+8x+12=0
(x+2)(x+6)=0
解得：x=-2或x=-6

当(x+3)(x+5)＝5时
x^2+8x+15=5
x^2+8x+10=0
根据求根公式得：
x=根号6 -4 或 -根号6 -4

Answer 2

(x+1)(x+3)(x+5)(x+7)+15 =[(x+1)(x+7)][(x+3)(x+5)]+15 =(x^2+8x+7)(x^2+8x+15)+15 =(x^2+8x+7)[(x^2+8x+7)+8]+15 =(x^2+8x+7)^2+8(x^2+8x+7)+15 =[(x^2+8x+7)+3][(x^2+8x+7)+5] =(x^2+8x+10)(x^2+8x+12) =(x^2+8x+10)(x+2)(x+6)

Answer 3

（x+1)(x+3)(x+5)(x+7)+15
=(x+1)(x+7)(x+3)(x+5)+15
=(x^2+8x+7)(x^2+8x+15)+15
=(x^2+8x)^2+22(x^2+8x)+105+15
=(x^2+8x)^2+22(x^2+8x)+120
=(x^2+8x+10)(x^2+8x+12)
=(x^2+8x+10)(x+2)(x+6)