We are given that the function $g(x) = ax^2 - 4x + 12$ and that $g(2) = -4$. The goal is to find the value of $g(3)$.

Step 1: Find the value of $a$

We know that $g(2) = -4$. Using this information, we substitute $x = 2$ into the function $g(x)$ and solve for $a$:

$g(2) = a(2)^2 - 4(2) + 12 = -4$ $4a - 8 + 12 = -4$ $4a + 4 = -4$ $4a = -8$ $a = -2$

Step 2: Substitute $a = -2$ into $g(x)$

Now that we know $a = -2$, substitute it into the function:

$g(x) = -2x^2 - 4x + 12$

Step 3: Calculate $g(3)$

Now, substitute $x = 3$ into the function:

$g(3) = -2(3)^2 - 4(3) + 12$ $g(3) = -2(9) - 12 + 12$ $g(3) = -18 - 12 + 12$ $g(3) = -18$

Thus, the value of $g(3)$ is $\boxed{-18}$, which corresponds to option C.

Would you like more details or have any other questions?

Here are five related questions to expand your understanding:

1. How would the answer change if $g(2) = 0$ instead of $g(2) = -4$?
2. Can you generalize how to solve for $a$ in any quadratic function if given a value for $g(x)$?
3. What is the vertex of the quadratic function $g(x) = -2x^2 - 4x + 12$?
4. How do we find the roots of the function $g(x)$, i.e., where $g(x) = 0$?
5. How does changing the coefficient $a$ affect the graph of a quadratic function?

Tip: When solving quadratic functions with unknown coefficients, use known function values to solve for those coefficients first!