Express f(t) in terms of unit step functions, then find the Laplace

I've been trying to do this problem forever. I'm not sure how to type this so I took a screenshot of the problem:

http://img85.imageshack.us/img85/3122/prob9.jpg

I have no idea how to do this. From what I could gather in my notes, I started off like this:

For the unit step functions of f(t):

t^2 - t^2U(t-1)

(t^2 - 1)U(t-1) - (t^2 - 1)U(t-2)

(t^2 - 1)U(t-2)

These may not be right... and even if they are right, can someone briefly explain why? I still don't have a thorough understanding of this.