How to factor this expression?

Question

Factor: $$x^{5}+a x^{3}+b x^{2}+cx+d$$ While: $$d^{2}+c b^{2}=abd$$ Please help me with this question.

I don't have any clue.

EDIT: I saw this but it was not explained what do we do after substitution. I still need help.

Answer 1

\begin{align*} x^5+ax^3+bx^2+cx+d &= x^5+ax^3+bx^2+\frac{d(ab-d)}{b^2}x+d \\ &= x^5+ax^3+bx^2+\left( \frac{ad}{b}-\frac{d^2}{b^2} \right)x+d \\ &= x^5-\frac{d^2x}{b^2}+ax\left( x^2+\frac{d}{b} \right)+(bx^2+d) \\ &= x\left( x^4-\frac{d^2}{b^2} \right)+ax\left( x^2+\frac{d}{b} \right)+ (bx^2+d) \\ &= x\left( x^2+\frac{d}{b} \right) \left( x^2-\frac{d}{b} \right)+ ax\left( x^2+\frac{d}{b} \right)+b\left( x^2+\frac{d}{b} \right) \\ &= \left( x^2+\frac{d}{b} \right) \left[ x\left( x^2-\frac{d}{b} \right)+ax+b \right] \\ &= \left( x^2+\frac{d}{b} \right) \left[ x^3+\left( a-\frac{d}{b} \right)x+b \right] \end{align*}