I have a [hypothetical] cubic equation in positive integers $n,a,b,c$, which is perhaps most compactly presented as $$ 2(2n-a)(2n-b)(2n-c) = 4n^2 - 2\bigl((a-b)^2+a+1\bigr)n + \bigl((a-b)^2c + b\bigr). $$
By hypothesis, each factor on the left-hand side is positive. I’m trying to prove that there are an infinite number of positive integers $n$ for which corresponding integers $a,b,c$ cannot be found to satisfy the equation.
Any hints or references on similar problems would be appreciated.