Find $(x + 1)^{−3}$ in $\mathbb{Z}_2[x]/(x^5 + x^4 + 1)$ In a previous question, I established that $x^5 + x^4 + 1$ is irreducible in $\mathbb{Z}_2[x]/(x^5 + x^4 + 1)$, so it will not be divisible by $x+1$. Hence $x^5+x^4+1$ will be coprime to $(x+1)^3 = x^3+x^2+x+1$ 
So far I have used the Euclidean algorithm to get:
$x^5+x^4+1 = (x^3+x^2+x+1)(x^2+1) + x$
$x^3+x^2+x+1 = (x^2+1)(x+1) + 0$
However, I know I have done something wrong - because the Euclidean algorithm should show that the gcd of $x^5+x^4+1$ and $x^3+x^2+x+1$ is $1$, not $(x^2+1)$.

Using the algorithm detailed in this answer with polynomials (and writing top to bottom instead of left to right),we get
$$
\begin{array}{c|c|c|c}
x^5+x^4+1&0&1&\\
x^3+x^2+x+1&1&0&\\
x&x^2+1&1&x^2+1\\
1&\color{#C00}{x^4+x^3+x}&x^2+x+1&x^2+x+1\\
0&x^5+x^4+1&x^3+x^2+x+1&x
\end{array}
$$
which says that in $\mathbb{Z}_2$,
$$
(x^3+x^2+x+1)^{-1}=x^4+x^3+x\pmod{x^5+x^4+1}
$$
