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The problem you're working on involves solving the inequality:

\[
(x+4)(x-8) \leq 0
\]

### Step 1: Identify critical points

The critical points of the inequality occur when the product equals zero:

\[
(x+4)(x-8) = 0
\]

The solutions to this equation are:
- \( x + 4 = 0 \) gives \( x = -4 \)
- \( x - 8 = 0 \) gives \( x = 8 \)

So, the critical points are \( x = -4 \) and \( x = 8 \).

### Step 2: Analyze intervals

The critical points divide the number line into three intervals:
1. \( (-\infty, -4) \)
2. \( (-4, 8) \)
3. \( (8, \infty) \)

Now, test a point from each interval in the inequality \( (x+4)(x-8) \leq 0 \):

- For \( x = -5 \) (in \( (-\infty, -4) \)): 
  \( (x+4)(x-8) = (-5+4)(-5-8) = (-1)(-13) = 13 \) (positive)
  
- For \( x = 0 \) (in \( (-4, 8) \)): 
  \( (x+4)(x-8) = (0+4)(0-8) = (4)(-8) = -32 \) (negative)
  
- For \( x = 9 \) (in \( (8, \infty) \)): 
  \( (x+4)(x-8) = (9+4)(9-8) = (13)(1) = 13 \) (positive)

### Step 3: Conclusion on sign

- The inequality \( (x+4)(x-8) \leq 0 \) is **negative or zero** in the interval \( [-4, 8] \).
- It is positive outside this interval.

Thus, the solution is:

\[
x \in [-4, 8]
\]

### Step 4: Plot the solution

- The solution includes \( x = -4 \) and \( x = 8 \) (since the inequality is \( \leq 0 \)).
- On the number line, you would have a closed interval from \( -4 \) to \( 8 \), including both endpoints.

### The plot on the number line
The graph you have already depicts this solution correctly. You see that the interval from \( -4 \) to \( 8 \) is highlighted, which corresponds to the solution \( [-4, 8] \).

Would you like a further breakdown of any part of the process? 

### Related Questions:
1. How do you solve quadratic inequalities using a sign chart?
2. Can you explain the difference between strict inequalities and non-strict inequalities?
3. What happens when the factors in the inequality have multiplicities greater than 1?
4. How does interval notation change

This problem focuses on solving the quadratic inequality (x+4)(x-8) ≤ 0 using interval analysis and factoring techniques. Learn how to break down the quadratic inequality, determine critical points, and graph the solution on a number line. Suitable for high school students, Grades 9-11.

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