I'm having trouble calculating the inverse of a polynomial. Consider the polynomial $f(x)=x^3+3x-2$, which is irreducible over $\mathbb{Q}$, as it has no rational roots. So $\mathbb{Q}[x]/(f(x))\simeq \mathbb{Q}[u]$ is a field.
How would I calculate $(u^2-u+4)^{-1}$ as a polynomial? Calling this $p(u)$, I tried solving for it by writing $ p(x)(x^2-x+4)=q(x)f(x)+1 $ but I don't know what $q(x)$ is. How can I write down $(u^2-u+4)^{-1}$ explicitly? Thanks.