#### Answer

$f(x) = -9x^{2} - 5x + 11$

#### Work Step by Step

We must first solve for the functions in terms of a, b, and c:
f(-2) = 4a - 2b + c = -15
f(-1) = a - b + c = 7
f(1) = a + b + c = -3
We can then form the matrix, and use Gaussian elimination and back-substitution to solve the matrix:
$\begin{bmatrix}
4 & -2 & 1 & |-15\\
1 & -1 & 1 & |7\\
1 & 1 & 1 & |-3\\
\end{bmatrix}$ ~ $\begin{bmatrix}
4 & -2 & 1 & |-15\\
0 & 2 & -3 & |-43\\
0 & -6 & -3 & |-3\\
\end{bmatrix}$ ~ $\begin{bmatrix}
4 & -2 & 1 & |-15\\
0 & 2 & -3 & |-43\\
0 & 0 & -12 & |-132\\
\end{bmatrix}$
C:
-12c = -132
c = 11
B:
2b - 3(11) = -43
2b - 33 = -43
2b = -10
b = -5
A:
4a - 2(-5) + 11 = -15
4a + 10 + 11 = -15
4a + 21 = -15
4a = -36
a = -9
Using the solution above, the quadratic function is:
$f(x) = -9x^{2} - 5x + 11$