https://file.aiquickdraw.com/mathbot/file/17286446149362vxl7u1f.jpg

Let's analyze the graph and the given quadratic inequalities to find which one corresponds to the solution set shown.

### Step-by-step Breakdown:

1. The graph shows a solution set on the number line between \(x = -2\) and \(x = 4\), with open circles at both points, indicating that these endpoints are excluded. Therefore, the solution corresponds to:
   \[
   -2 < x < 4
   \]
   
2. Now, let's examine each quadratic expression and test if it matches the graph:
   
   #### Option 1: \(x^2 - 2x - 8 < 0\)
   We can factor this expression to check its roots:
   \[
   x^2 - 2x - 8 = (x - 4)(x + 2)
   \]
   The roots are \(x = 4\) and \(x = -2\). So the quadratic inequality becomes:
   \[
   (x - 4)(x + 2) < 0
   \]
   This inequality is satisfied for values between \(x = -2\) and \(x = 4\), excluding the endpoints. This matches the solution set shown in the graph.

   #### Option 2: \(x^2 + 2x - 8 < 0\)
   Let's factor this:
   \[
   x^2 + 2x - 8 = (x - 2)(x + 4)
   \]
   The roots are \(x = 2\) and \(x = -4\), which doesn't match the graph since the solution set would be between \(x = -4\) and \(x = 2\).

   #### Option 3: \(x^2 - 2x - 8 > 0\)
   This is the opposite inequality of Option 1. It represents values outside the interval \(x = -2\) to \(x = 4\), so this doesn't match the graph.

   #### Option 4: \(x^2 + 2x - 8 > 0\)
   Similar to Option 2, this would represent values outside the interval \(x = -4\) to \(x = 2\), which also doesn't match the graph.

### Conclusion:
The inequality that represents the solution set shown in the graph is:
\[
\boxed{x^2 - 2x - 8 < 0}
\]

Would you like further details on the solution or explanations about any specific step?

Here are 5 related questions to expand on this:
1. How can you solve quadratic inequalities algebraically?
2. What is the graphical interpretation of quadratic inequalities?
3. What do open and closed circles on a number line indicate?
4. How does the factorization of a quadratic equation help in solving inequalities?
5. What does it mean when the inequality sign is reversed?

**Tip:** Always check the behavior of a quadratic expression by factoring and identifying the intervals where it satisfies the inequality.

The graph below represents the solution set of which inequality? Options: x² - 2x - 8 < 0, x² + 2x - 8 < 0, x² - 2x - 8 > 0, x² + 2x - 8 > 0.

Learn how to solve the quadratic inequality x² - 2x - 8 < 0 using graphical methods. Understand factoring, inequality rules, and how to interpret number line solutions with open endpoints. This problem is suitable for students in Grades 9-11.

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