$\frac{2}{n^2+5n} \geq \frac{2}{n}$ for all $n \geq 1$, and $\sum \frac{2}{n}$ converges.

$\frac{2}{n^2+5n} \leq \frac{2}{n^2}$ for all $n \geq 1$, and $\sum \frac{2}{n^2}$ converges.

$\frac{2}{n^2+5n} \leq \frac{2}{n}$ for all $n \geq 1$, and $\sum \frac{2}{n}$ converges.

$\frac{2}{n^2+5n} = \frac{2}{n(n+5)}$ and therefore the series diverges.