Integrating by parts we obtain...
(1)
... where we use the fact that if exists, then is...
(2)
Kind regards
for the inverse transform use partial fractions to split the expression up. you can check if i did it correctly but i got (1/2)/(s-3) + (1/2)/(s+1). so you should be able to recognize these as laplace transforms of the exponential function. so you should get (1/2)e^(3t) + (1/2)e^(-t).