(a) Suppose is a solution for this recurrence, with . Then
So we expect a general solution of the form
With , we get four equations in four unknowns:
So the particular solution to the recurrence is
(b) Let be the generating function for . Multiply both sides of the recurrence by and sum over all .
From here you would write each term as a power series (easy enough, since they're all geometric or derived from a geometric series), combine the series into one, and the solution to the recurrence will be the coefficient of , ideally matching the solution found in part (a).