We want to examine whether the ideal $I=(X^2+4,X)$ is a maximal ideal of $\Bbb Z[X]$, which as we know is not a PID.
This result tells us precisely which are the maximal ideals of $\Bbb Z[X]$. Thus it seems that this is not maximal, but I can't see why if so. It might be easy but I feel I stuck here.
Any help is appreciated, thanks.
The first way is something more sophisticated. Recall that an ideal $I \subset R$ is a maximal ideal iff $R/I$ is a field. In this case, note that we have $$\frac{\Bbb Z[X]}{(X^2 + 4, X)} \cong \frac{\Bbb Z}{(4)}.$$ The latter is not a field and thus, $(X^2 + 4, X)$ is not maximal.
The other way is to explicitly produce a larger proper ideal. To see this, one can note that $$(X^2 + 4, X) \subsetneq (X^2 + 4, X, 2) \subsetneq \Bbb Z[X].$$ (Why are the inclusions proper?)