$9\mathrm{.}45.$ Consider the LTI system shown in Figure P$9\mathrm{.}45($a$)$ for which we are given the following information:

$x(s)=\frac{s+2}{s-2}$

$x(t)=0,t>0,$

and

$y(t)=-\frac{2}{2}{e}^{2t}u(-t)+\frac{1}{2}{e}^{-t}u(t).,[$ See Figure $P9\mathrm{.}45(b).]$

$($a$)$ Determine $H(s)$ and its region of convergence.

$($b$)$ Determine $h(t).$

$($c$)$ Using the system function $H($s$)$ found in part $($a$),$ determine the output $y(t)$ if the input is

$x(t)=e^{3t},-$