Basis for the vector space P2 I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial 
$
ax^2 + bx + c
$
with the matrix 
$
A = \begin{bmatrix}
1,0,0 \\
0,1,0 \\
0,0,1 \\
\end{bmatrix}
$
and the vector 
$
\begin{bmatrix}
1 \\
x \\
x^2 \\
\end{bmatrix}
$
what corresponds to $a$, $b$, and $c$ in the matrix $A$?

I think you need to be clear about what you mean by "representing" the polynomial.  You can if you like make the assignments
$$
x^2 \;\; \to \;\; \left [ \begin{array}{c}
0\\
0\\
1\\
\end{array} \right ] \hspace{2pc} x \;\; \to \;\; \left [ \begin{array}{c}
0\\
1\\
0\\
\end{array} \right ] \hspace{2pc} 1 \;\; \to \;\; \left [ \begin{array}{c}
1\\
0\\
0\\
\end{array} \right ]
$$
Then your polynomial can be represented by the vector
$$
ax^2 + bx + c \;\; \to\;\; \left [ \begin{array}{c}
c\\
b\\
a\\
\end{array} \right ].
$$
To describe a linear transformation in terms of matrices it might be worth it to start with a mapping $T:P_2 \to P_2$ first and then find the matrix representation.
Edit: To answer the question you posted, I would take each basis vector listed above and apply the matrix to it:
\begin{eqnarray*}
\left [ \begin{array}{ccc}
3 & 2 & 7 \\
0 & 1 & 0 \\
4 & 0 & 1 \\
\end{array} \right ] \left [ \begin{array}{c}
1 \\
0\\
0\\
\end{array} \right ] & = & \left [ \begin{array}{c}
3 \\
0 \\
4 \\
\end{array} \right ] \;\; \to \;\; 4x^2 + 3 \\
\left [ \begin{array}{ccc}
3 & 2 & 7 \\
0 & 1 & 0 \\
4 & 0 & 1 \\
\end{array} \right ] \left [ \begin{array}{c}
0 \\
1\\
0\\
\end{array} \right ] & = & \left [ \begin{array}{c}
2 \\
1\\
0\\
\end{array} \right ] \;\; \to \;\; x+ 2 \\
\left [ \begin{array}{ccc}
3 & 2 & 7 \\
0 & 1 & 0 \\
4 & 0 & 1 \\
\end{array} \right ] \left [ \begin{array}{c}
0 \\
0\\
1\\
\end{array} \right ] & = & \left [ \begin{array}{c}
7 \\
0\\
1\\
\end{array} \right ] \;\; \to \;\; x^2 + 7.
\end{eqnarray*}
