Given,

$\rm \frac{x^2}{y ( x- y)} + \frac{y^2}{x (y - x)}$

We know, $\rm (y - x) = - (x - y)$

$\rm = \frac{x^2}{y (x - y)} + \frac{y^2}{x \{ - (x-y) \}}$

$\rm = \frac{x^2}{y (x-y)} - \frac{y^2}{x (x -y)}$

Taking LCM and simplifying the equation, we get,

$\rm = \frac{ x^2 \cdot x - y^2 \cdot y}{xy (x - y)}$

$\rm = \frac{x^3 - y^3}{xy (x - y)}$

Using the factorization formula for $\rm (a^3 - b^3) = (a-b)(a^2 + ab + b^2)$, we get,

$\rm = \frac{ (x - y)(x^2 + xy + y^2)}{xy (x -y)}$

$\rm = \frac{ x^2 + xy + y^2}{xy}$

$$\rm \therefore \frac{x^2}{y (x - y)} + \frac{y^2}{x (y -x)} = \frac{x^2 + xy + y^2}{xy}$$