Solution
= Solution
The transform factors as $s^{-2}(s+2)^{-2}$, whose inverse factors are $t$ and $te^{-2t}$. The <convolution theorem>[Laplace convolution theorem] gives
$$
f(t)=\int_0^t\tau(t-\tau)e^{-2(t-\tau)},d\tau,
$$
and direct integration produces
$$
f(t)=\frac14\bigl(t-1+(t+1)e^{-2t}\bigr),
$$
confirming part (b).