Proving iff statement about polynomials I have the following question:
Let $\mathbb F$ be a field and $ x^3 +ax+b, x^2+cx-1 $ $\in \mathbb F$[x]. 
Prove that $(x^2+cx-1) | (x^3+ax+b) $if and only if $c=b$ and $ a= -1-b^2$. 
Attempt:
If $(x^2+cx-1) | (x^3+ax+b) $ then there exists $(x+d)$ such that $(x^2+cx-1)(x+d) = (x^3+ax+b) $. Then I got $ (c+d)x^2 + (cd-1)x -d=ax+b$ and I don't know how to do the next step. Can anyone help me?

Just equate the coefficients and eliminate $d$. We have $c+d=0, cd-1=a$ and $-d=b$. So $d=-c$. Plug this into the other equations to get $a=cd-1=-1-c^{2}$ and $b=-d=c$. Hence $b=c$ and $a =-1-b^{2}$. Now check that the converse is also true. 
