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

$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$

where $a = -1$ and $b = 4$.

First, calculate $g(-1)$:

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

Next, calculate $g(4)$:

$g(4) = (4)^3 - 2 = 64 - 2 = 62$

Now, substitute these values into the formula:

$\text{Average rate of change} = \frac{62 - (-3)}{4 - (-1)} = \frac{62 + 3}{4 + 1} = \frac{65}{5} = 13$

So, the average rate of change is 13.