Our strategy is to determine the values of the linear transformation on a “nice” basis for the domain, and then apply the ideas of Theorem LTDB to obtain our formula. $\vectrepname{B}$ is a linear transformation of the form $\ltdefn{\vectrepname{B}}{P_3}{\complex{4}}$, so for a basis of the domain we choose a very simple one: $C=\set{1,\,x,\,x^2,\,x^3}$. We now give the vector representations of the elements of $C$, which are obtained by solving the relevant systems of equations obtained from linear combinations of the elements of $B$.
\begin{align*}
\vectrep{B}{1} &=\colvector{20 \\ 7 \\ 1 \\ 4}
&
\vectrep{B}{x} &=\colvector{17 \\ 14 \\ -8 \\ -3}
&
\vectrep{B}{x^2}&=\colvector{-3 \\ -3 \\ 2 \\ 1}
&
\vectrep{B}{x^3}&=\colvector{0 \\ -1 \\ 1 \\ 1}
&
\end{align*}
This is enough information to determine the linear transformation uniquely, and in particular, to allow us to use Theorem LTLC to construct a formula.
\begin{align*}
\vectrep{B}{a+bx+cx^2+dx^3} &=
a\vectrep{B}{1} + b\vectrep{B}{x} + c\vectrep{B}{x^2} + d\vectrep{B}{x^3}\\
&=
a\colvector{20 \\ 7 \\ 1 \\ 4} +
b\colvector{17 \\ 14 \\ -8 \\ -3} +
c\colvector{-3 \\ -3 \\ 2 \\ 1} +
d\colvector{0 \\ -1 \\ 1 \\ 1}\\
&=
\colvector{
20a + 17b - 3c\\
7a + 14b - 3c - d\\
a - 8b + 2c + d\\
4a - 3b + c + d
}
\end{align*}