To find the average rate of change of the function $h(x) = x^2 + 7x + 6$ over an interval, we use the formula for the average rate of change:

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

Where $a$ and $b$ are the endpoints of the interval over which you're calculating the average rate of change.

So, for a general interval $[a, b]$, we substitute $h(a)$ and $h(b)$ into the formula.

Step-by-Step:

1. Find $h(a)$: Substitute $a$ into the function $h(x) = x^2 + 7x + 6$. $h(a) = a^2 + 7a + 6$

2. Find $h(b)$: Substitute $b$ into the function $h(x)$. $h(b) = b^2 + 7b + 6$

3. Calculate the average rate of change: $\text{Average rate of change} = \frac{(b^2 + 7b + 6) - (a^2 + 7a + 6)}{b - a}$ Simplify the expression in the numerator: $\text{Average rate of change} = \frac{b^2 + 7b + 6 - a^2 - 7a - 6}{b - a} = \frac{b^2 - a^2 + 7b - 7a}{b - a}$

4. Factor the numerator: The difference of squares $b^2 - a^2$ factors as: $(b^2 - a^2) = (b - a)(b + a)$ So, the formula becomes: $\text{Average rate of change} = \frac{(b - a)(b + a) + 7(b - a)}{b - a}$

5. Cancel out $b - a$ from the numerator and denominator: $\text{Average rate of change} = b + a + 7$

Thus, the average rate of change of $h(x) = x^2 + 7x + 6$ over the interval $[a, b]$ is:

$\boxed{b + a + 7}$

Would you like to evaluate this for specific values of $a$ and $b$, or do you have any questions about the process?

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Additional Questions:

1. What is the instantaneous rate of change of $h(x)$ at a specific point?
2. How does the concept of average rate of change relate to the derivative of a function?
3. Can the average rate of change ever be negative for a quadratic function like $h(x)$?
4. What would the graph of $h(x) = x^2 + 7x + 6$ look like?
5. How do you compute the slope of a tangent line to the curve at a given point?

Tip: The average rate of change gives the overall change in the function's output relative to its input, but it doesn’t provide information about how the function behaves at any specific point within the interval.