To find the average rate of change of the function $g(x) = -2x^3 - 4$ from $x = -1$ to $x = 4$, we will use the formula for the average rate of change:

$\text{Average Rate of Change} = \frac{g(4) - g(-1)}{4 - (-1)}$

Step 1: Find $g(4)$

$g(4) = -2(4)^3 - 4 = -2(64) - 4 = -128 - 4 = -132$

Step 2: Find $g(-1)$

$g(-1) = -2(-1)^3 - 4 = -2(-1) - 4 = 2 - 4 = -2$

Step 3: Calculate the average rate of change

$\text{Average Rate of Change} = \frac{g(4) - g(-1)}{4 - (-1)} = \frac{-132 - (-2)}{4 - (-1)} = \frac{-132 + 2}{5} = \frac{-130}{5} = -26$

So, the average rate of change is $-26$.

Would you like further clarification on any part of this? Here are some related questions you might find interesting:

1. What does the average rate of change represent graphically?
2. How does this concept compare to the derivative at a point?
3. How would the rate of change change if the interval was different?
4. Can the average rate of change ever be positive for this cubic function?
5. How would you interpret the result $-26$ in a real-world context?

Tip: The average rate of change is essentially the slope of the secant line between two points on the graph of the function over the given interval.