let $J$ be an ideal of $F[x]$ such that $J$ contains every polynomial of $\mathbb{Z}[x]$ with constant term a multiple of 4. Im looking for a ring isomorphic to $\mathbb{Z}[x]/J$, with proof.

I was thinking that maybe $\mathbb{Z}[x]/J \cong \mathbb{Z}\_4[x]$,

I started by defining a homomorphism $\phi : \mathbb{Z}[x] \to \mathbb{Z}\_4[x]$, where $\phi(a(x)) = \overline{a(x)}$ for any $a(x) \in \mathbb{Z}[x]$