Compute the convolution \((f * g)(t)\) (in the Laplace transform sense) and its corresponding Laplace transform \(\mathcal{L}[f * g]\) for the following functions:

a. \(f(t)=t^{2}, g(t)=t^{3}\).

b. \(f(t)=t^{2}, g(t)=\cos 2 t\).

c. \(f(t)=3 t^{2}-2 t+1, g(t)=e^{-3 t}\).

d. \(f(t)=\delta\left(t-\frac{\pi}{4}\right), g(t)=\sin 5 t\).