Let $F$ be integer domain but not be field, then every ideal of $F[x]$ is prime ideal but not maximal?.
Now, I set a homomorphism $\varphi: R[x]\to R$ defined by $\varphi(a_0+a_1 x+\ldots+a_n x^n)=a_0$
then by First isomorphism theorem: $R[x]/\ker{\varphi}\cong R$
and $\ker{\varphi}=a_0+a_1 x+\ldots +a_n x^n$ such that $\varphi(a_0+a_1 x+\ldots+a_n x^n)=0$, or $a_0=0$, so $\ker{\varphi}=\langle x \rangle$. Because $R$ is integer domain, $R[x]/\langle x \rangle$ also is integer domain. That is $\langle x \rangle$ is prime ideal.
But why this is not maximal?.
In particular $R=\mathbb{Z}$, then $\langle x \rangle$ is not maximal because $\langle x \rangle \subset \langle 2,x \rangle$.
Is in general we also have $\langle x \rangle \subset \langle 2,x \rangle$?.