To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x^{2}+y^{2}$ and $\left(x+y\right)\left(x-y\right)$ is $\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)$. Multiply $\frac{x^{2}}{x^{2}+y^{2}}$ times $\frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}$. Multiply $\frac{x^{2}}{\left(x+y\right)\left(x-y\right)}$ times $\frac{x^{2}+y^{2}}{x^{2}+y^{2}}$.
Since $\frac{x^{2}\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ and $\frac{x^{2}\left(x^{2}+y^{2}\right)}{\left(x+y\right)\left(x-y\right)\left(x^{2}+y^{2}\right)}$ have the same denominator, add them by adding their numerators.
