I believe you need to re-write the piecewise function in terms of the unit step function.
Here is the question:
Screenshot by Lightshot
Basically what I've done is change it into:
s^{2}Y - sy_{0} - y'_{0 }+ 4sY - 4y_{0}+ 3Y = F(s)
Which I'm not even sure is right, can anyone point me in the right direction now?