jmc

Call the two integers $x$ and $y$. Without loss of generality, let $x$ be the larger of the two. We are given that $x+y=50$ and $x-y=12$, and we are asked for $x^2-y^2$. Because $x^2-y^2$ factors into $(x+y)(x-y)$, we can simply substitute in to get $x^2-y^2=50\cdot 12=\boxed{600}$.