What rule is used for this simplification?

Question

$$ \frac{8}{(s+1)^2 + 2^2} \times \frac{1}{s} = \frac{8}{5} - \frac{1}{s} + \frac{16}{10}\times \frac{s+1}{(s+1)^2 + 2^2} + \frac{8}{10}\times \frac{2}{(s+1)^2 + 2^2} $$

Answer 1

Do some partial fraction magic?

$$\frac{8}{s(s^2+2s+5)} = \frac{As+B}{s^2+2s+5} + \frac{C}{s}$$

This gives $C=8/5, A=-8/5, B=-16/5$ I think, so

$$\frac{8}{s(s^2+2s+5)} = \left(\frac{8}{5}\right)\left[\frac{1}{s}-\frac{s+2}{s^2+2s+5}\right].$$