Homework 7 collaboration area

What exactly is 6.2 #9 asking when it says to use another method to find the laplace transform for Prob 1? (AM, 07-Oct)

When checking my answer for 6.3 #8, the inverse laplace transform of my solution is correct for positive s however goes on for (1-e^-t) toward -infinity. Is this ok since the laplace transform only cares about positive time? Or is my solution missing something? (AM, 07-Oct)