From Förberedande kurs i matematik 1

If we multiply the top and bottom of the first fraction by \displaystyle x+5 and the second by \displaystyle x+3, then they will both have the same denominator and we can work out the expression by subtracting the numerators

\displaystyle \begin{align} \frac{2}{x+3}-\frac{2}{x+5} &= \frac{2}{x+3}\cdot \frac{x+5}{x+5}-\frac{2}{x+5}\cdot \frac{x+3}{x+3}\\[7pt] &= \frac{2(x+5)-2(x+3)}{(x+3)(x+5)}\\[5pt] &= \frac{2x+10-2x-6}{(x+3)(x+5)}\\[5pt] &= \frac{4}{(x+3)(x+5)}\,\textrm{.} \end{align}