(How to show that something isn't an) Irreducible Polynomial in $\mathbb{Z}_5$ 
Let $f(x) = x^4+x+p$, where $p$ is a prime. I'm asked to show that if $p \neq -1$ (mod $5$), then $f(x)$ is not irreducible in $\mathbb{Z}_5$.

I guess I could try to show that $f(x)$ is not irreducible for $p=0,1,2,3$, but that seems like a lot of work and I'm guessing there's a more efficient way which I don't know.

It's not that much work:
$$\begin{cases}
p\equiv 0 \pmod 5 & 0\text{ is a root}\\
p\equiv 1 \pmod 5 & 3\text{ is a root}\\
p\equiv 2 \pmod 5 & 2\text{ is a root}\\
p\equiv 3 \pmod 5 & 1\text{ is a root}
\end{cases}$$
