To solve for the average rate of change of the function $h(x) = 2x^2 - 7x$ from $x = 3$ to $x = 5$, we can use the following formula for the average rate of change:

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

where $a = 3$, $b = 5$, and $h(x) = 2x^2 - 7x$.

Step-by-step solution:

1. First, calculate $h(3)$ and $h(5)$.

   $h(3) = 2(3)^2 - 7(3) = 2(9) - 21 = 18 - 21 = -3$

   $h(5) = 2(5)^2 - 7(5) = 2(25) - 35 = 50 - 35 = 15$

2. Now, apply the values in the formula for average rate of change:

   $\text{Average rate of change} = \frac{h(5) - h(3)}{5 - 3} = \frac{15 - (-3)}{5 - 3} = \frac{15 + 3}{2} = \frac{18}{2} = 9$

So, the average rate of change of $h(x)$ from $x = 3$ to $x = 5$ is $9$.

Would you like more details, or have any other questions?

Here are 5 related questions for further understanding:

1. How is the average rate of change related to the slope of the secant line between two points?
2. What is the difference between average rate of change and instantaneous rate of change?
3. How would the average rate of change differ if the interval was $x = 2$ to $x = 5$?
4. Can the average rate of change be negative? What does that indicate about the function?
5. How would you calculate the derivative of $h(x)$, and what does it represent?

Tip: The average rate of change of a function over an interval is essentially the slope of the line connecting the function's values at the two endpoints of that interval.