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