Laplace Transform of this signal?

In my linear systems class, we are doing Laplace transforms using transform tables and the properties. I can usually do the problems that closely resemble the table, but when they involve heavy algebraic manipulations, I don't know what to do.

Can someone explain the steps of this solution?

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Why do the +1 and -1 appear? Where did the extra exponential come from? If someone can go line for line, I would appreciate it and how to in general solve these kinds of signals?

Re: Laplace Transform of this signal?