Actually we can prove the statement in the reverse way.
Here $a$ is non zero element, $F$ is a field.
Suppose $f(x)$ is reducible then $f(x) = g(x)h(x)$, then $f(x+a)= g(x+a) h(x+a)$.
But I can't understand what it exactly wants to convey.
It's an exercise problem from Joseph Gallion Algebra under Factorization of Polynomials.