Để phân tích đa thức thành nhân tử, ta cần áp dụng công thức khai triển đa thức và rút gọn các thành phần tương tự.

3) (x^2+4x+8)^2+3x(x^2+4x+8)+2x^2
= (x^2+4x+8)(x^2+4x+8) + 3x(x^2+4x+8) + 2x^2
= (x^2+4x+8)(x^2+4x+8 + 3x + 2)
= (x^2+4x+8)(x^2+7x+16)

4) (x^2+x+1)^2+5x(x^2+x+1)+x^2
= (x^2+x+1)(x^2+x+1) + 5x(x^2+x+1) + x^2
= (x^2+x+1)(x^2+x+1 + 5x + 1)
= (x^2+x+1)(x^2+6x+2)

5) (x^2-x+1)^2-5x(x^2-x+1)+4x^2
= (x^2-x+1)(x^2-x+1) - 5x(x^2-x+1) + 4x^2
= (x^2-x+1)(x^2-x+1 - 5x + 4)
= (x^2-x+1)(x^2-6x+5)

6) (x^2-x+2)^4-3x^2(x^2-x+2)^2+2x^4
= (x^2-x+2)(x^2-x+2)(x^2-x+2)(x^2-x+2) - 3x^2(x^2-x+2)(x^2-x+2) + 2x^4
= (x^2-x+2)(x^2-x+2)(x^2-x+2)(x^2-x+2 - 3x^2(x^2-x+2) + 2x^4
= (x^2-x+2)(x^2-x+2)(x^2-x+2 - 3x^4 + 3x^3 - 6x^2 + 4x^2 - 4x + 8x - 8 + 2x^4
= (x^2-x+2)(x^2-x+2)(x^2 - 3x^4 + 3x^3 - 2x^2 + 12x - 8)

7) 3(-x^2+2x+3)^4-26x^2(-x^2+2x+3)^2-9x
= 3(-x^2+2x+3)(-x^2+2x+3)(-x^2+2x+3)(-x^2+2x+3) - 26x^2(-x^2+2x+3)(-x^2+2x+3) - 9x
= 3(-x^2+2x+3)(-x^2+2x+3)(-x^2+2x+3)(-x^2+2x+3 - 26x^2(-x^2+2x+3) - 9x
= 3(-x^2+2x+3)(-x^2+2x+3)(-x^2+2x+3 - 26x^4 + 52x^3 - 78x^2 - 26x^4 + 52x^3 - 78x^2 + 39x^2 - 78x + 117x - 117 - 9x
= 3(-x^2+2x+3)(-x^2+2x+3)(-52x^4 + 104x^3 - 117x^2 + 48x - 126)

8) -6(-x^2-x+1)^4+x^2(-x^2-x+1)^2+5x^4
= -6(-x^2-x+1)(-x^2-x+1)(-x^2-x+1)(-x^2-x+1) + x^2(-x^2-x+1)(-x^2-x+1) + 5x^4
= -6(-x^2-x+1)(-x^2-x+1)(-x^2-x+1)(-x^2-x+1) + x^2(-x^2-x+1)(-x^2-x+1) + 5x^4
= -6(-x^2-x+1)(-x^2-x+1)(-x^2-x+1)(-x^2-x+1 - x^4 + 2x^3 - x^2 - x^4 + 2x^3 - x^2 + x^2 - 2x + x - x^2 + 2x - 1 + 5x^4
= -6(-x^2-x+1)(-x^2-x+1)(-x^2-x+1 - 2x^4 + 4x^3 - 3x^2 + 2x - 1 + 5x^4
= -6(-x^2-x+1)(-x^2-x+1)(-7x^4 + 4x^3 - 2x^2 + 2x - 1)