Hmm. I'm not sure I buy your derivation there. Here's the original DE:
The LT of the DE is
or
Solving for I get
I think this step was incorrect:
It should be
I am having trouble with the problem attached, I need to find the inverse laplace of X(s).
I can do the 3rd term from my table of conversions directly, the 2nd term I have an answer for using partial fractions (although i'm not 100% it's correct). The 1st term is my biggest problem though.
The answer to aim for is: x(t) = 1 - e^(2t)