with initial conditions of and

Partial Fraction Decomposition:

Equating like Coefficients:

Solving the system:

New Functions to take the inverse Laplace transform of:

Breaking down further:

Note:Completed the square on above

Now my problem is the middle term, I don't have any table entry at all for something like that, I have table entries for the first term and third term though. How would I break down the middle term further? Would really appreciate it if someone could help me finish this up, as it's been stumping me for a while now.

The answer in the book is:

Is all of my work thus far correct though?