Answer
See below
Work Step by Step
We are given: $f(t)=t^2\\
g(t)=e^t$
Obtain: $(f * g)=\int^t_0f(t-x)g(x)dx\\
=\int^t_0 e^{x} (t-x)^2 dx\\
=\int^t_0 e^{x}(t^2-2tx+x^2) dx\\
=t^2\int^t_0 e^x dx-2t \int^t_0 xe^x dx+\int^t_0 x^2e^x dx\\
=\begin{bmatrix}
t^2e^x -2t(xe^x-e^x)+x^2e^x-2xe^x+2e^x
\end{bmatrix}^t_0\\
=2e^t-t^2-2t-2$
