First note that $y = a + bx + cx^2$ can also be written as
$(1 \quad x \quad x^2) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = y$
You are given three points, $(x_1,y_1) = (1,0)$, $(x_2,y_2) = (-1,-4)$ and $(x_3,y_3) = (2,11)$. So you can set up three of these equations:
$\begin{align} (1 \quad x_1 \quad x_1^2) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) &= y_1 \\ (1 \quad x_2 \quad x_2^2) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) &= y_2 \\ (1 \quad x_3 \quad x_3^2) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) &= y_3 \end{align}$
Or, in one matrix form:
$\left(\begin{array}{ccc} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ 1 & x_3 & x_3^2 \end{array}\right) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = \left(\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right)$
You know the values of $x_1, x_2, x_3, y_1, y_2, y_3$, so filling them in you will be left with an equation of the form $Ax = b$, with $A$ and $b$ known and $x$ unknown.
Edit: For completeness, filling in those values we get
$\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 2 & 4 \end{array}\right) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = \left(\begin{array}{c} 0 \\ -4 \\ 11 \end{array}\right)$
You could do a Gaussian elimination, but this example has a simple solution which we can just calculate by hand. Subtracting the second row from the first we get
$(0 \quad 2 \quad 0) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = 4 \quad \Longrightarrow \quad \fbox{b = 2}$
Subtracting the first from the last we get
$(0 \quad 1 \quad 3) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = 11 \quad \Longrightarrow \quad b + 3c = 11 \quad \Longrightarrow \quad \fbox{c = 3}$
Then finally you could use the first row to get
$(1 \quad 1 \quad 1) \left(\begin{array}{c} a \\ b \\ c \end{array}\right) = 0 \quad \Longrightarrow \quad a + b + c = 0 \quad \Longrightarrow \quad \fbox{a = -5}$
(For the enthusiast, note that the matrix $A$ is a Vandermonde matrix.)