Partial fraction decomposition find the inverse laplace transform

Question

It may sound as a silly question but I want to know clearly why the following partial fraction decomposition is wrong: Suppose we're solving a ODE using Laplace transform, we find that the Laplace transform of our solution is $F(s)=\frac{1}{s^2(s+1)}$, and that we write it as $F(s)=\frac{A}{s^2}+ \frac{B}{s+1}$, thus finding that the solution is $y(t)=At+Be^{-t}$, which is obviously wrong because we forgot the term $C/s$ in the partial fraction decomposition. The question is, why is it wrong, even if we chose A and B such that $\frac{A}{s^2}+ \frac{B}{s+1}=\frac{1}{s^2(s+1)}$?