Your partial fractions are wrong.

It should be A/s + B/(s + 3) = 1/(s^2 + 3s)

[A(s + 3) + Bs]/[s(s + 3)] = 1/[s(s + 3)]

A(s + 3) + Bs = 1

As + 3A + Bs = 1

(A + B)s + 3A = 0s + 1

A + B = 0 and 3A = 1

A = 1/3 and B = -1/3.

So 1/(s^2 + 3s) = 1/(3s) - 1/[3(s + 3)]. Now find the inverse Laplace transform.