To determine the average rate of change of the function $g(x) = x^2 + 10x + 23$ over the interval $[-9, 1]$, we use the formula:

$\text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}$

where $a = -9$ and $b = 1$.

1. First, calculate $g(-9)$: $g(-9) = (-9)^2 + 10(-9) + 23 = 81 - 90 + 23 = 14$

2. Next, calculate $g(1)$: $g(1) = (1)^2 + 10(1) + 23 = 1 + 10 + 23 = 34$

3. Now, find the average rate of change: $\text{Average Rate of Change} = \frac{g(1) - g(-9)}{1 - (-9)} = \frac{34 - 14}{1 + 9} = \frac{20}{10} = 2$

The average rate of change of the function $g(x)$ over the interval $[-9, 1]$ is 2.

Would you like more details on any of the steps, or do you have any questions?

Here are some related questions you might find interesting:

1. How do you find the instantaneous rate of change at a specific point for a quadratic function?
2. What is the geometric interpretation of the average rate of change on a graph?
3. How does the average rate of change relate to the slope of the secant line?
4. Can the average rate of change be negative? What does it imply?
5. How does the average rate of change differ between linear and quadratic functions?

Tip: Always remember to substitute the values of $x$ carefully when calculating function outputs, as small errors can lead to incorrect results.