Apply L to both sides:

(s^2 Y(s) - 1s - 0) + 3(s Y(s) - 1) + 2 Y(s) = 1/s^2.

Solve for Y(s):

(s^2 + 3s + 2) Y(s) = (s + 3) + 1/s^2 = (s^3 + 3s^2 + 1)/s^2.

==> Y(s) = (s^3 + 3s^2 + 1)/(s^2 (s^2 + 3s + 2))

By partial fractions,

Y(s) = (1/4) [2/s^2 - 3/s + 12/(s + 1) - 5/(s + 2)].

Inverting term by term:

Y(s) = (1/4) (2t - 3 + 12e^(-s) - 5e^(-2t)].

I hope this helps!