Factor out a root from this polynomial

Question

Let a be a root of $f$, ie. $ a^3 + a^2 = -1, f = x^3 + x^2 + 1 = (x-a)(x^2+..??)$

I tried this but got $f=(x-a)(x^2 + 2ax + 2a^2)$ which obviously doesn't work

Answer 1

Hint. If $a^3 + a^2 = -1$ then $$x^3+x+1=(x-a)(x^2+Ax+B)=x^3+(A-a)x^2+(B-aA)x-aB$$ which implies that $A=a+1$ and $B=a+a^2$. However this not helpful for the factorization...

Answer 2

The answer is $f(x)=(x-a)(x^2+bx+c)=x^3+(b-a)x^2+(c-ab)x-ca$ which is implies that: \begin{align} b-a=1\\ c-ab=0\\ -ca=1 \end{align} The above equations can be solved for $b$ and $c$. However, solution to these equations is conditioned by $a$ (not for all $a$ you can solve these equations for $b$ and $c$). Obviously, this is related to the fact that $a$ is a root of $f(x)=0$.