Suppose we want to prove whether $(X^3 + X) \subset \mathbb{R}[X]$ is prime and or maximal. I would go about like:
Pick $I = (X)$ en $J = (X^2 + 1)$. Then $I$ and $J$ are coprime since $-X \cdot X \in I$ en $X^2+1 \in J$ and $-X \cdot X + X^2 + 1 = -X^2 + X^2 + 1 = 1$. Thus by the CRT we have $$ \mathbb{R}[X]/(X^3 + X) \cong \mathbb{R}[X]/(X) \times \mathbb{R}[X]/(X^2 + 1) \cong \mathbb{R} \times \mathbb{C} $$ Since it is a product ring, it is neither a field nor a domain, so $(X^3 + X)$ is neither prime nor maximal.
However, I suddenly got confused about the specifics of the CRT. It says if $I$ and $J$ are two coprime ideals, then $$ R/(I \cdot J) \cong (R/I) \times (R/J) $$ But by our choice of $I$ and $J$, does it holds that $$ (I \cdot J) = ((X) \cdot (X^2+1)) = (X^3 + X)? $$ I can prove $\supseteq$ but not $\subseteq$, what is going wrong here?
First, note that $X,X^2+1\notin(X^3+X)$ (by degree considerations) but their product does belong to the ideal, so this ideal is not prime (and hence not maximal). Next, take $f(X)\in(X)(X^2+1)$ then $f(X)=g(X)·X·h(X)·(X^2+1)$ for some $g(X),h(X)\in\mathbb{R}[X]$ then you get $f(X)=g(X)h(X)·X(X^2+1)=g(X)h(X)·(X^3+X)\in(X^3+X)$. Hope this helped