TITLE: Image of an integer polynomial
QUESTION [5 upvotes]: How do I describe all integers that can be written in the form $$(x+y)^2+5x+3y$$ for some integers $x$ and $y$?

REPLY [3 votes]: Another approach is to rewrite it as $$(x+y)(x+y+3) + 2x.$$
This immediately shows us why every even number is possible, e.g. Vadim set $x+y = 0$, though we could also have $x+y+3 = 0$.
This also immediately shows us why every odd number is impossible, since both summands are even.