Using the theorem of convolution find the inverse Laplace transform

Question

Using the theorem of convolution find the inverse Laplace transform: \begin{align} F(s)=\frac{s}{(s-1)(s+2)}\end{align}

This is what I have so far:

\begin{align} Let \space F_1(s)=\frac{s}{s-1} and\space let\space F_2(s)=\frac{1}{(s+2)}\end{align}

Find the Laplace inverse of these function we would get:

\begin{align}L^{-1}(\frac{s}{s-1})=L^{-1}(1-\frac{1}{s-1})=\delta(t)-e^t \space and \end{align}

\begin{align} L^{-1}(\frac{1}{s+2})=e^{-2t} \end{align}

This is where I got stuck. I have tried multiple things and end up with the Dirac delta function $\delta(t)$, this is the first time I hear about this function and do not know how to integrate it.

My question, is there any other way to solve this problem without the Dirac delta function?