Correct Answer - Option 4 :
\(\frac{2s}{(s+2)^3}\)
Concept:
Output of a system Y(s) = Input X(s) × Transfer function H(s)
\(L\left\{ {{t^n}} \right\} = \frac{{n!}}{{{s^{n + 1}}}}\)
Analysis:
y(t) = t2 e2t
\(Y(s) = \frac{2}{(s+2)^3}\)
x(t) = u(t)
\(X(s) = \frac{1}{s}\)
Y(s) = X(s) × H(s)
\(H(s) = \frac{Y(s)}{X(s)}\)
\(= \frac{2s}{(s+2)^3}\)