Suppose $a_1, \dots, a_n$ are distinct integers and let
$$ f(x) = [(x - a_1) \cdots (x - a_n)]^4 + 1. $$
Show that $f(x)$ is irreducible over $\mathbb{Q}$.
For the case $f(x) = [(x - a_1) \cdots (x - a_n)]^2 + 1$, I supposed $f(x)$ was reducible ($f(x) = g(x)h(x)$) and got to a contradiction manipulating the degrees of $g(x)$ and $h(x)$, using a result concerning the $n$ values $a_1, \dots, a_n$ for which $f(x) = 1$. However, I tried to adapt this same argument here and it didn't work because now the degree of $f$ is higher ($4n$) and the contradiction doesn't arise immediately.
How should I proceed?
