Express \[\frac{2}{r^2+4r+3}\] in partial fractions.

Hence, or otherwise, prove that \[\sum_{r=1}^n \frac{2}{r^2+4r+3}=\frac{5}{6}-\frac{2n+5}{(n+2)(n+3)}.\]

We know how to rewrite the expression using partial fractions – maybe we can use this to simplify the sum?

Maybe we should try writing the sum out explicitly term by term?