Let $m, n ∈ \mathbb{N}$ such that $2m^{2} + m = 2n^{2} + n$ , then prove that $m-n$ is a perfect square . After simply factorizing the equation given in the question I got $\mathbf{m + n = -0.5}$
. But the question mentioned $m, n ∈ \mathbb{N}$ . Then how $m + n = -0.5$ ?
Did I do some mistake or is the question wrong?

Yes. $m-n$ is a perfect square.
Just observe that your condition implies that
$$ (m-n)(2m+2n+1)=0$$
Since $m$ and $n$ are natural numbers, we must have $m-n = 0$ which is a perfect square
