Answer
The matrix representatino of T relative to the basis $B=\{1,t,t^2\}$ is $\begin{bmatrix}
3&0&0\\
5&-2&0\\
0&4&1
\end{bmatrix}$
Work Step by Step
$T[a_0]=\begin{bmatrix}
3\\
5t\\
0
\end{bmatrix}$, $T[a_1]=\begin{bmatrix}
0\\
-2t\\
4t^2
\end{bmatrix}$, $T[a_2]=\begin{bmatrix}
0\\
0\\
t^2
\end{bmatrix}$
$T[a_0]_B=\begin{bmatrix}
3\\
5\\
0
\end{bmatrix}$, $T[a_1]_B=\begin{bmatrix}
0\\
-2\\
4
\end{bmatrix}$, $T[a_2]_B=\begin{bmatrix}
0\\
0\\
1
\end{bmatrix}$
The matrix representatino of T relative to the basis $B=\{1,t,t^2\}$ is
$\begin{bmatrix}
3&0&0\\
5&-2&0\\
0&4&1
\end{bmatrix}$
