Show that $I=\langle x+1, x^2+1\rangle$ is maximal in $\mathbb Z[x]$. I’m trying to show this by showing that $\mathbb Z[x]/I$ is a field. Since in this ring $x=-1$ and so $x^2=1$, but also $x^2=-1$ from which it follows that $1=-1$ and so it is probably isomorphic to $\mathbb F_2$? But I can’t find a homomorphism from $\mathbb Z[x]\ \longrightarrow\ \mathbb F_2$ with kernel $I$.

I would put it this way:
\begin{align}
\mathbf Z[X]\bigm/(X+1,X^2+1)&=\mathbf Z[X]\bigm/(X+1,X^2+1\bmod X +1))\\
&=\mathbf Z[X]\bigm/(X+1,2)\simeq\mathbf Z/2\mathbf Z[X]/(X+1)\\
&\simeq\mathbf Z/2\mathbf Z.
\end{align}
