partial fraction expansion:
I'll assume that you'll be able to find the constants a and b.
Now all we need to do is look at the transform table (I'll assume that the TF is casual) :
...
(5+s)/(s^2 + 2s)
= 5/(s^2 + 2s) + s/(s^2 + 2s)
= 5/s(s + 2) + s/s(s + 2)
= 5/s(s + 2) + 1/(s + 2)
applying partial fraction to the first term we get,
= 5/2s -5/2(s+2) + 1/(s+2)
applying inverse Laplace transform (using the table)
=5/2 -5 exp(-2t)/2 + exp(-2t)