To solve the inequality \( x^2 + 2x - 8 < 0 \), let's follow these steps:

1. **Find the roots of the corresponding equation** \( x^2 + 2x - 8 = 0 \).

   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -8 \):

   \[
   x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-8)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm \sqrt{36}}{2}
   \]

   \[
   x = \frac{-2 \pm 6}{2}
   \]

   So, the roots are:

   \[
   x = \frac{-2 + 6}{2} = 2 \quad \text{and} \quad x = \frac{-2 - 6}{2} = -4
   \]

2. **Determine the intervals defined by these roots:**

   The roots divide the number line into three intervals:

   - \( (-\infty, -4) \)
   - \( (-4, 2) \)
   - \( (2, \infty) \)

3. **Test each interval to see where the inequality is true:**

   - For \( (-\infty, -4) \): Choose \( x = -5 \)
     \[
     (-5)^2 + 2(-5) - 8 = 25 - 10 - 8 = 7 > 0
     \]

   - For \( (-4, 2) \): Choose \( x = 0 \)
     \[
     (0)^2 + 2(0) - 8 = -8 < 0
     \]

   - For \( (2, \infty) \): Choose \( x = 3 \)
     \[
     (3)^2 + 2(3) - 8 = 9 + 6 - 8 = 7 > 0
     \]

4. **Conclusion:**

   The inequality \( x^2 + 2x - 8 < 0 \) is true for the interval \( (-4, 2) \).

Therefore, the solution is:

\[
-4 < x < 2
\]

Would you like more details or have any questions?

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Here are 5 related questions to consider:

1. How would the solution change if the inequality was \( x^2 + 2x - 8 \leq 0 \)?
2. What is the graph of \( y = x^2 + 2x - 8 \), and how does it relate to the inequality solution?
3. Can you solve a similar inequality like \( x^2 - 4x + 3 < 0 \)?
4. How do you determine the nature of the roots of a quadratic equation without solving it?
5. What are the properties of quadratic inequalities in terms of concavity and vertex?

**Tip:** When solving inequalities, always check intervals by selecting test points to determine where the inequality holds true.

This guide explains how to solve the quadratic inequality x^2 + 2x - 8 < 0 with a detailed step-by-step solution. The solution involves finding roots, testing intervals, and determining where the inequality holds. Key concepts include quadratic equations and interval analysis. Suitable for Grades 8-10.

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