There is a quick trick to solving for the coefficients of this partial fraction. To find A for example I multiply both sides of the equation by (s+3), now evaluate the left hand side at s = -3 since B is gone.
For A i get (s-3)/(s-1) evaluated at -3 gives 6/4 or 3/2. For B i do the same but multiply both sides by (s-1) and get -1/2
So now take the inverse laplace transform by using a look up table if you wish and I get 1.5e^(-3t) - 0.5e^(-t)