Finding the laplace transform of this function (Looks like piecewise functions)

........ {0 if t < 5
f(t) = {t² if 5 < t < 8
........ {0 if 8 < t

Rewrite in terms of step functions:
f(t) = t^2 u(t - 5) - t^2 u(t - 8)
.....= [(t - 5) + 5]^2 u(t - 5) - [(t - 8) + 8]^2 u(t - 8)
.....= [(t - 5)^2 + 10(t - 5) + 25] u(t - 5) - [(t - 8)^2 + 16(t - 8) + 64] u(t - 8).

So, L{f(t)} = [2/s^3 + 10/s^2 + 25/s] e^(-5s) - [2/s^3 + 16/s^2 + 64/s] e^(-8s).

I hope this helps!