Question

A voltage v(t)=2t^(2)e^(-3t) is passed through a delay line having a delay of 4 sec.Detrmine Laplace transform of the output voltage.

Answer 1

I think your question was: \[v(t) = 2t ^{2} e ^{-3t}\]
You need to apply the following rules:
\[L[u(t)] = 1/s\]
\[L[e ^{at} f(t)] = F(s - a)\]
\[L[t ^{n} f(t)] = (-1)^{n}F ^{n}(s)\]
The Laplace transform is:\[ V(s) = 4/(s + 3)^{3}\]

Now to account for the delay. The delay in time manifests as a phase shift in s-domain. So an additional phase shifting term e^(-as) joins the result. So the final result after the 4s delay is:
\[ V(s) = (4 e ^{-4s})/(s + 3)^{3}\]