Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.

If $(2m+1)^2-4(2n+1)$ is a square, then it has the form $q^2$ where $q$ is an odd number (since $(2m+1)^2-4(2n+1)$  is odd). In this case, the quadratic equation
$$x^2-(2m+1)x+(2n+1)=0$$
has a pair of solutions which are integers (since $q$ is odd).
But both the sum $(2m+1)$ and the product $(2n+1)$ of such solutions are odd. This is a contradiction (since for arbitrary integers $a$ and $b$, at least one of $a+b$ and $ab$ must be even).
