Math Help - LaPlace Transformations to Solve Ordinary Differential Equations

Hey guys I am trying to do the following problem (with attempted solution):

LaPlace Transformations to Solve Ordinary Differential Equations

Any help/advice would be much appreciated. I am really stumped on this one

So you're really asking how to do the inverse Laplace Transform, right? I agree with your work until that point, incidentally.

I would complete the square in the denominator, to start with. Then you could use some of the shifting theorems to handle the exponential. After that, I think you could essentially just write down the answer.