non-uniqueness of factorization in $\Bbb{Z}_{12}[x]$

Question

I just finished a unit on the uniqueness of factorization in $F[x]$, but my textbook didn't give me an example of non-uniqueness in $R[x]$ where $R$ isn't a field. Is there a simple example of a polynomial in $\Bbb{Z}_{12}[x]$ (or some other non-field) that has multiple factorizations into irreducibles? In other words, I know why the proof depends on $F$ being a field, I just would like to see an example showing why uniqueness doesn't work in $R[x]$. Thanks!

Answer 1

I would go with $$(x-1)(x+1)=x^2-1=x^2-25=(x-5)(x+5).$$