few questions with laplace transform

I am having trouble this this problem.

Heres the work I have up until where I get stuck:

partial fractions:

How do I solve this partial fraction equation now? I have tried it and I am ending up with a different answer than the solutions manual for B.

My other question is:

Find the fundeamental set of solutions using only real coefficients:

I get the steps:

I am lost as to where you get is equal to and how this works on a graph....

Thanks for the help guys, I really appreciate it!