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 e^{2t}

\(Y(s) = \frac{2}{(s+2)^3}\)

x(t) = u(t)

\(X(s) = \frac{1}{s}\)

Y(s) = X(s) × H(s)

\(H(s) = \frac{Y(s)}{X(s)}\)

\(= \frac{2s}{(s+2)^3}\)