Hey.
How can I solve this problem using Convolution?
Do you have to use convolution? Why not break it up into partial fractions ....?
$\displaystyle \frac{5s}{(s^2 + 9) (s - 4)} = \frac{as+b}{s^2 + 9} + \frac{c}{s-4} = a \, \frac{s}{s^2 + 3^2} + \frac{b}{3} \, \frac{3}{s^2 + 3^2} + \frac{c}{s + (-4)}$ ....
If you must use convolutions, I'd note that $\displaystyle \frac{5s}{(s^2 + 9) (s - 4)} = 5 \, \frac{s}{s^2 + 3^2} \cdot \frac{1}{s + (-4)}$ and apply the convolution theorem ......
Mr Fantastic appears to know what you are talking about, but I do not see a problem just a Laplace transform or transfer function. So what are we being asked to solve using convolution?
If you want to find the function that this is the LT of using the convolution theorem you need to know that this is it or equation 10 here.
RonL