Answer
Matrix for $T$ relative to $B$ and $C$ is $\begin{bmatrix}
5&0&0\\
1&5&0\\
0&1&5\\
0&0&1\\
\end{bmatrix}$
Work Step by Step
a) $(t+5)(2-t+t^2)=t^3+4t^2-3t+10$
b) For any $\vec{p}$ and $\vec{q}$ in $P_2$ and any scalar k,
$T[\vec{p}+\vec{q}]=(t+5)[\vec{p}+\vec{q}]=(t+5)\vec{p}+(t+5)\vec{q}=T[\vec{p}]+T[\vec{q}]$
$T[k\vec{p}]=(t+5)(k\vec{p})=k(t+5)[\vec{p}]=kT\vec{p}$
c) Let $B=\{1,t,t^2\}$ and $C=\{1,t,t^2, t^3\}$
$T[b_1]=t+5$
$T[b_2]=t^2+5t$
$T[b_3]=t^3+5t^2$
$T[b_1]_C=\begin{bmatrix}
5\\
1\\
0\\
0\\
\end{bmatrix}$, $T[b_2]_C=\begin{bmatrix}
0\\
5\\
1\\
0\\
\end{bmatrix}$, and $T[b_3]_C=\begin{bmatrix}
0\\
0\\
5\\
1\\
\end{bmatrix}$
Matrix for $T$ relative to $B$ and $C$ is
$\begin{bmatrix}
5&0&0\\
1&5&0\\
0&1&5\\
0&0&1\\
\end{bmatrix}$
