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