ISL 1989 Problem 51

Let $f(x) = \prod^n_{k=1} (x - a_k) - 2,$ where $n \geq 3$ and $a_1, a_2, \ldots,$ an are distinct integers. Suppose that $f(x) = g(x)h(x),$ where $g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $n = 3.$