How to solve the simple inverse Laplace transform of $\frac{-(s+5)}{5(s+1)}$.

Question

I haven't touched Laplace transforms in quite some time so I'm very rusty.

I'm trying to reverse:

$$ \frac{-(s+5)}{5(s+1)} = \frac{-s}{5(s+1)} + \frac{1}{(s+1)}$$

Obviously, the second term is $ e^{-t} $ in the time domain. However, I'm unable to figure out what the process would be to solve the first term?

Is it possible I'm forgetting some Laplace rule of the multiplication of two functions?

Answer 1

I'd suggest to write it as $$ -\frac15 \left[ 1 + \frac{4}{s + 1} \right] $$

Answer 2

Hint:

\begin{align*} -\frac{s+5}{5(s+1)}&=\frac{-(s+1)-4}{5(s+1)}\\ &=-\frac{s+1}{5(s+1)}-\frac{4}{5(s+1)}\\ &=-\frac{1}{5}-\frac{4}{5}\frac{1}{s+1} \end{align*}