demonstrate the resolution of laplace inverses, I got a slightly scary result, but I wanted to clear up a doubt.
$\mathcal{L}^{-1}\left\{ \dfrac{1}{s^2(s+2)} \right\} - \mathcal{L}^{-1}\left\{ \dfrac{e^{-s}}{s^2(s+2)}\right\} - \mathcal{L}^{-1}\left\{\dfrac{e^{-s}}{s(s+2)}\right\} $
In the first inverse of Laplace, I set up a system using separation of attainment, obtaining a=-1/4, b=1/2, and c=1/4. getting, then, after the inverse of laplace, with the result equal to $\frac{1}{4}*4H(t) + \frac{1}{2}*t + \frac{1}{4} * e^{-2t}$