For $n>1$, let $a_1, a_2, \dots, a_n$ be $n$ distinct integers. Prove that the polynomial $$f(x)=(x-a_1)(x-a_2)...(x-a_n) - 1$$ cannot be written as the product of two nonconstant polynomials with integer coefficients.
My Proof (or attempt)
Assume that $f(x)$ can be written as $h(x)\cdot g(x)$. Note for any $x$, $f(x)$ must be prime. This means either $h(x)$ or $g(x)$ must equal $1$, but since these polynomials must be non-constant, we have a contradiction, and we are done.
Source: Art of Problem Solving Vol. 2