To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2x+1$ and $2x-1$ is $\left(2x-1\right)\left(2x+1\right)$. Multiply $\frac{2}{2x+1}$ times $\frac{2x-1}{2x-1}$. Multiply $\frac{2}{2x-1}$ times $\frac{2x+1}{2x+1}$.
Since $\frac{2\left(2x-1\right)}{\left(2x-1\right)\left(2x+1\right)}$ and $\frac{2\left(2x+1\right)}{\left(2x-1\right)\left(2x+1\right)}$ have the same denominator, add them by adding their numerators.
