factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$

Question

I want to factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$ but I don't know how to use the second equality.I tried a lot but I cannot know how to use it for example it is $d^2$ but we have $d$ and if we calculate $d$ using that equality it would be harder.Any hints?

Answer 1

Substituting $c=(abd-d^2)/b^2$ we obtain the factorization $$ x^5+ax^3+bx^2+cx+d=\frac{(abx + b^2 + bx^3 - dx)(bx^2 + d)}{b^2}. $$ This is the way I would use the second equality.