Let's expand the left side of the equation:

$x-y$^2 = x^2 - 2xy + y^2
$x+y$^2 = x^2 + 2xy + y^2

Adding these two expressions together, we get:

$x^2-2xy+y^2$ + $x^2+2xy+y^2$ = 2x^2 + 2y^2 = 2$x^2+y^2$

Therefore, $x-y$^2 + $x+y$^2 = 2$x^2+y^2$