https://file.aiquickdraw.com/mathbot/file/1727711304702s834q1eq.jpg

Let's solve the quadratic inequality given in the image:

\[
x^2 - 2x - 8 < 0
\]

### Step 1: Factor the quadratic expression
We need to factor \(x^2 - 2x - 8\). To factor this expression, find two numbers that multiply to \(-8\) and add up to \(-2\). These numbers are \(-4\) and \(2\). So, the factored form is:

\[
(x - 4)(x + 2) < 0
\]

### Step 2: Find the critical points
The critical points (where the expression equals 0) are found by setting each factor equal to 0:

\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

### Step 3: Test the intervals
The critical points split the number line into three intervals: \((- \infty, -2)\), \((-2, 4)\), and \((4, \infty)\).

Now we test a point from each interval to determine where the expression is negative:

- **For \(x = -3\) (in \((- \infty, -2)\)):**
  \[
  (x - 4)(x + 2) = (-3 - 4)(-3 + 2) = (-7)(-1) = 7 > 0
  \]
  This interval does not satisfy the inequality.

- **For \(x = 0\) (in \((-2, 4)\)):**
  \[
  (x - 4)(x + 2) = (0 - 4)(0 + 2) = (-4)(2) = -8 < 0
  \]
  This interval satisfies the inequality.

- **For \(x = 5\) (in \((4, \infty)\)):**
  \[
  (x - 4)(x + 2) = (5 - 4)(5 + 2) = (1)(7) = 7 > 0
  \]
  This interval does not satisfy the inequality.

### Step 4: Conclusion
The inequality \( (x - 4)(x + 2) < 0 \) holds true for \(x \in (-2, 4)\).

Thus, the solution to the inequality is:

\[
(-2, 4)
\]

This matches the **blue answer** in the quiz.

---

Would you like further explanation or details on any steps? Here are five related questions you can explore:

1. How do you solve quadratic equations using the quadratic formula?
2. What are the differences between solving quadratic inequalities and quadratic equations?
3. How do you graph the solution to a quadratic inequality on a number line?
4. What are the roles of the critical points in solving inequalities?
5. How can you solve inequalities with higher powers like cubic inequalities?

**Tip:** When solving inequalities, always test values in different intervals to determine where the inequality holds true!

Solve the quadratic inequality: x^2 - 2x - 8 < 0

Learn how to solve the quadratic inequality x^2 - 2x - 8 < 0 using factoring and testing intervals. This step-by-step solution explores factoring methods and the use of critical points to find the correct intervals. Suitable for students in Grades 9-11.

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