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Convolution Applied to Inverse Transform Problem

Please see the attached file for the fully formatted problem.

My problem lies in manipulating the equation to one which the inverse transform can be taken, but would appreciate this example of convolution worked with some more of the blanks filled. I cannot figure out how the equation
gets manipulated into what appears to be .
The main objective here is to find the inverse Laplace transform for h(s) using convolution.

k and t are separate variables, not a single variable kt.

Side Note: Is there a mathematical method for taking the inverse transform or are there just tables from people who did the original Laplace transform?

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