1. **State the problem:** Find the average rate of change of the function $g(x) = 2x^3 + 2$ from $x = -2$ to $x = 3$. 2. **Formula:** The average rate of change of a function $g(x)$ over the interval $[a, b]$ is given by $$\text{Average rate of change} = \frac{g(b) - g(a)}{b - a}$$ where $a = -2$ and $b = 3$. 3. **Calculate $g(-2)$:** $$g(-2) = 2(-2)^3 + 2 = 2(-8) + 2 = -16 + 2 = -14$$ 4. **Calculate $g(3)$:** $$g(3) = 2(3)^3 + 2 = 2(27) + 2 = 54 + 2 = 56$$ 5. **Apply the formula:** $$\frac{g(3) - g(-2)}{3 - (-2)} = \frac{56 - (-14)}{3 + 2} = \frac{56 + 14}{5} = \frac{70}{5}$$ 6. **Simplify the fraction:** $$\frac{\cancel{70}}{\cancel{5}} = 14$$ 7. **Final answer:** The average rate of change of $g(x)$ from $x = -2$ to $x = 3$ is $14$.