I'm having trouble understanding the process behind the Extended Euclidean Algorithm. I know that $\mathbb{Q}[x]/(x^3-2)$ is a field with the greatest common divisor of $1+x$ and $x^3-2$ being 1 since $x^3-2$ is irreducible over $\mathbb{Q}$, so there exists polynomials $r(x)$ and $s(x)$ such that $$1 = r(x)(x+1)+s(x)(x^3-2).$$Since $x^3 \equiv 2 \,(\text{mod } x^3-2)$, if we substitute $x^3$ for 2 we have $$1 = r(x)(x+1) + 0$$ so if we find $r(x)$, we find the inverse of $(x+1)$. I understand that we can use the extended euclidean algorithm to find a linear combination of our two polynomials $x+1$ and $x^3-2$, but I don't understand the process. Here's what I have
\begin{align} x^3-2 &= (x+1)(x^2-x+1)+(-3) \\ x+1 &= (-3)\left(\frac{-1}{3}x-\frac{1}{3}\right)+0 \end{align} How do I use this information to find $r(x)$?
