Exhibit a reducible polynomial of the form $x^p -x-c$ having no roots in a field of characteristic 0

Question

Is it possible for a polynomial, $x^p -x-c$ where $p$ is prime, to be reducible in a field of characteristic $0$, yet have roots in that field?

I know for a fact that the general form is true, that a polynomial of degree 2 or 3 in $F[x]$ is irreducible if and only if it has no roots in $F$. Thus, I am looking at $p=5$ as the simplest counterexample.