Laplace Transform using initial conditions

Question

I have

w" + w = t^2 + 2 with w(0) = 1, and w'(0) = -1

I was able to figure out that w(t) = t^2 + cos(t) - sin(t)

but I cannot figure out what W(s) is

I thought it was = (2/s^3) + 2/s but it says I am incorrect.

Answer 1

s²Ws) - s(1) + 1 + W(s) = 2/s³ + 2/s; W(s) = (2/s³ + 2/s + s - 1)/(s² + 1) = (s⁴ - s³ + 2s² + 2)/[s³(s² + 1)]

W(s) = 2/s³ + (s - 1)/(s² + 1);

w(t) = L^-1(W(s) = t² + cost - sint.

I use partial fraction decomposition.