The average rate of change of a function between two points is given by the formula:

$\frac{g(b) - g(a)}{b - a}$

where $g(x)$ is the function, and $a$ and $b$ are the two given points.

Given $g(x) = -4x^2 + 5x + 4$, $a = -2$, and $b = 2$, we need to:

1. Calculate $g(-2)$
2. Calculate $g(2)$
3. Find the difference $g(2) - g(-2)$
4. Divide the result by $2 - (-2) = 4$

Step 1: Calculate $g(-2)$

$g(-2) = -4(-2)^2 + 5(-2) + 4 = -4(4) - 10 + 4 = -16 - 10 + 4 = -22$

Step 2: Calculate $g(2)$

$g(2) = -4(2)^2 + 5(2) + 4 = -4(4) + 10 + 4 = -16 + 10 + 4 = -2$

Step 3: Find $g(2) - g(-2)$

$g(2) - g(-2) = -2 - (-22) = -2 + 22 = 20$

Step 4: Divide by $b - a = 2 - (-2) = 4$

$\frac{g(2) - g(-2)}{2 - (-2)} = \frac{20}{4} = 5$

Thus, the average rate of change of $g(x)$ between $x = -2$ and $x = 2$ is:

$\boxed{5}$

Would you like more details on any step or have any questions?

Here are 5 related questions to expand on this:

1. What is the interpretation of the average rate of change in the context of functions?
2. How would you calculate the instantaneous rate of change at a specific point?
3. Can the average rate of change be negative? If so, what does that imply?
4. How is the average rate of change related to the slope of a secant line?
5. How would the average rate of change differ if we were given different points?

Tip: The average rate of change is similar to finding the slope between two points on a graph of the function.