I have tried as below.

To find inverse of Laplace transform, I want to make partial fraction as below.

$$\begin{array}{r}{\displaystyle \frac{2}{s({s}^{2}+4)}}={\displaystyle \frac{A}{s}}+{\displaystyle \frac{Bs+C}{{s}^{2}+4}}={\displaystyle \frac{(A+B){s}^{2}+Cs+4A}{s({s}^{2}+4)}}.\end{array}$$

After that, we have system of linear equation

$$\begin{array}{rl}A+B& =0\\ C& =0\\ 4A& =2.\end{array}$$

Thus we have A=2, B=−2, and C=0. Now, substituting A,B,C and we have

$$\begin{array}{r}{\displaystyle \frac{2}{s({s}^{2}+4)}}={\displaystyle \frac{2}{s}}+{\displaystyle \frac{-2s}{{s}^{2}+4}}.\end{array}$$

But the fact is

$$\begin{array}{r}{\displaystyle \frac{2}{s({s}^{2}+4)}}\ne {\displaystyle \frac{2}{s}}+{\displaystyle \frac{-2s}{{s}^{2}+4}}={\displaystyle \frac{8}{{s}^{2}+4}}.\end{array}$$

I'm stuck here. I can't make a partial fraction for F(s) and I can't find inverse of Laplace transform for F(s).

Anyone can give me hint to give me hint for this problem?