Certainly! I'll walk you through each step of the solution in the image you provided. This is a detailed process for finding the slope (denoted by $m$) of the secant line between two points on a curve.

Step-by-step Explanation:

1. Given Points: The points provided are $(-4, 56)$ and $(-4 + h, k(-4 + h))$. This indicates two points on the graph, with one point being fixed at $(-4, 56)$ and the other point shifting based on the variable $h$.

2. Slope Formula: The slope $m$ is calculated as: $m = \frac{k(-4+h) - 56}{(-4+h) - (-4)}$ This is the standard slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, where the second point is $(-4+h, k(-4+h))$ and the first point is $(-4, 56)$.

3. Simplifying the Denominator: The denominator simplifies as: $(-4+h) - (-4) = h$ So the slope equation becomes: $m = \frac{k(-4+h) - 56}{h}$

4. Substituting the Function $k(x)$: Based on the problem setup, $k(x) = 4x^2 + 2x$. So, for $k(-4+h)$, we substitute into this function: $k(-4+h) = 4(-4+h)^2 + 2(-4+h)$

5. Expanding the Expression $k(-4+h)$: First, expand $(-4+h)^2$: $(-4+h)^2 = 16 - 8h + h^2$ Now, substitute this into the function $k(-4+h)$: $k(-4+h) = 4(16 - 8h + h^2) + 2(-4+h)$ Simplify the terms: $k(-4+h) = 64 - 32h + 4h^2 - 8 + 2h = 56 - 30h + 4h^2$

6. Substituting Back into the Slope Formula: Now, substitute $k(-4+h) = 56 - 30h + 4h^2$ into the slope formula: $m = \frac{(56 - 30h + 4h^2) - 56}{h}$ This simplifies to: $m = \frac{-30h + 4h^2}{h}$

7. Simplifying the Slope Expression: Factor out $h$ from the numerator: $m = \frac{h(-30 + 4h)}{h}$ The $h$'s cancel out: $m = -30 + 4h$

Thus, the final expression for the slope $m$ is: $m = -30 + 4h$

Summary of Steps:

1. Start with two points $(-4, 56)$ and $(-4+h, k(-4+h))$.
2. Use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
3. Simplify the denominator to $h$.
4. Substitute the function $k(x) = 4x^2 + 2x$ and expand for $k(-4+h)$.
5. Simplify and factor the expression.
6. Cancel out $h$ to get the final slope.

Do you want more details on any of these steps, or do you have any questions?

Here are five related questions:

1. What is the geometric meaning of the slope of a secant line?
2. How would this process change if the function $k(x)$ were different?
3. What is the limit of the slope $m$ as $h \to 0$?
4. How does this process relate to the concept of the derivative?
5. Can this method be applied to find the slope of a tangent line?

Tip: When simplifying algebraic expressions, be careful to expand and combine like terms correctly before factoring.