consider some function like $y = x^2 + 3x$
and then some family of related polynomial functions
(like: $y = x^2 + 3x$, $y = 2x^2 + 4x$, $y = 3x^2 + 5x$, $y = 4x^2 + 6x$, etc.)
what method or set of tools would conventionally be used for determining/defining all integers $y$ that CANNOT be produced by any integer $x$ in any one of these functions?
I understand that the study of Diophantine equations seems sort of relevant, but it also seems like it focuses more on the opposite (the solutions, not the $y$'s that can't be solutions), and in any event I don't understand what specific methods it would offer to tackle this kind of problem.