Let $E=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of the equation $x^3+x^2+x+2=0.$
Express $(\alpha^2+\alpha+1)(\alpha^2+\alpha)$ and $(\alpha-1)^{-1}$ in the form $a\alpha^2+b\alpha+c$ with $a,b,c\in \mathbb{Q}$.
My approach: Firstly, note that $\alpha$ should be irrational by rational root test. The first one can be done easily in the following way:
$(\alpha^2+\alpha+1)(\alpha^2+\alpha)=\alpha^2(\alpha^2+\alpha+1)+\alpha(\alpha^2+\alpha+1)=-2\alpha-2.$
But I don't know how to solve the second one. I have tried different approaches but none of them give something useful.
Would be very grateful for any help!
If you divide the polynomial $x^{3} + x^{2} + x + 2$ by $x-1$, you may get something like $$ x^{3} + x^{2} + x + 2 = (x-1)Q(x) + R. $$ Now put $x = \alpha$, which gives $0 = (\alpha-1)Q(\alpha) + R$. Now you may see how to proceed after.