Using complex numbers for 2nd order DE

I can change this equation, using Eulers method, and get the answers using and , but using complex numbers is apparently faster and we have to do it this way for my professor...so if you could help me figure this out I would appreciate it.

HERE IS WHAT I DID:

First, I'll solve the homogeneous case to find the complementary solution and the roots.

or

Now, to find the solution , where is the particular solution to the non-homogeneous equation, I need help. I guessed a few functions but none seemed to work out.

Here is what Ive tried:

Since ...well the real part I think

and

is a root of the homogenous equatoin, so I need multiply my guess for by

From here I cannot figure out what function I need to use...