I'm trying to solve the following exercise:
Show that for $\alpha,\beta\geq 3$, the polynomial $f = X(X-3)(X-\alpha)(X-\beta) + 1\in\mathbb Z[X]$ is irreducible.
It is straightforward to check that the polynomial $f$ doesn't have any rational roots. So the only remaining possibility is a decomposition into $2$ factors of degree $2$. Here, I don't know how to proceed.
The factorization $$ X(X-1)(X-2)(X-3) + 1 = (X^2 - 3X + 1)^2 $$ shows that it won't be possible to get a contradiction via the reduction modulo any number $n$.