Does $I\subset R$ maximal ideal implies $IR[x]\subset R[x]$ maximal ideal?

Question

Let $R$ be a ring and $I$ be an ideal of $R$. Then define $$IR[x]=\{f=\sum a_i x^i\in R[x] \;|\; a_i\in I\}$$

I can prove that if $I$ is prime then so is $IR[x]$, what about maximal ideal correspondence?

I mean, if $I$ is maximal then I think we can't say $IR[x]$ is maximal.

Here is my approach:

Note that we can prove $$\frac R I[x]\cong \frac{R[x]}{IR[x]}$$

if $I$ is maximal then $R/I$ is field then $\frac R I[x]\cong \frac{R[x]}{IR[x]}$ is Euclidean Domain but not necesserily a field.

So $IR[x]$ might not be a maximal ideal.

Try to think for a long time but couldnot produce any counterexample.