Polynomial divisor in $\mathbb{Z}_p[x]$?

Question

Q: Find an odd prime $p$ for which $x-2$ is a divisor of $x^4 + x^3 + 3x^2 + x + 1$ in $\mathbb{Z}_p[x]$.

I would rather not go through the Euclidean Algorithm for every mod $p$. Is there another way to find this particular prime?

Any help would be greatly appreciated.