Given

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

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

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

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

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

Using the factor formula for $\rm (x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

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

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

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

Hence, the required simplified form of the above expression is found.