The question and answer is attached though I need some help understanding the following:

1) When finding the laplace transform of the function, how do you know which order to do it in? For example, I could do it the order given in the answer or I could transform this way:

First, I transform u(t) to 1/s

Then I transform $\displaystyle e^-at * sin wt$ to $\displaystyle w/(((s+a)^2)+w^2)$

Hence, multiplying the two transforms I get,

1/s . (w)/(s+a)^2 + w^2

Which is different from the given answer, so does the order matter and if so, how do u decide which order to do it in?

2) I think the solution is wrong because when completing the square i got $\displaystyle (s+3/2)^2+(7/4)$ rather than what was given. Is it possible the answer is wrong?

Thanks for your time!