We know from derivative of transforms $$\begin{array}{rl}{F}^{\prime }(s)& =-L(tf(t))\\ ⟹L\left(\frac{1}{2}t{e}^{-3t}\right)& =-\frac{1}{2}\frac{d}{ds}L\left(e^{-et}\right)\\ & =\frac{-1}{2}\frac{d}{ds}\frac{1}{s+3}\\ & =\frac{1}{2}\frac{1}{{(s+3)}^2}\end{array}$$