Is the following fraction (actually a Laplace transform) a kind of partial fraction?
$$\frac{4s+3}{{s^2}+3}$$
Can this be solved this way? $$\frac{A}{s}+\frac{B}{s+{\frac{3}{s}}}$$
If not can you please tell me how to find inverse transform?
If you want to keep everything real, this is already decomposed into partial fractions. For the inverse Laplace transform, just split it as $$ \frac{4s+3}{s^2+3} = 4 \frac{s}{s^2+3} + 3\frac{1}{s^2+3}. $$ You should be able to invert each term separately.
If you don't need to keep it real, the roots of $s^2 + 3$ are $\pm \sqrt{3} i$, and the partial fraction decomposition is $$\frac{4s+3}{s^2+3} = {\frac {2-i\sqrt {3}/2}{s-i\sqrt {3}}}+{\frac {2+i\sqrt {3}/2}{ s+i\sqrt {3}}}$$