To solve the inequality \(x^2 - 2x - 15 > 0\), we will first solve the corresponding equation \(x^2 - 2x - 15 = 0\) to find the boundary points, and then analyze the inequality.

### Step 1: Solve \(x^2 - 2x - 15 = 0\)
Factor the quadratic equation:

\[
x^2 - 2x - 15 = (x - 5)(x + 3) = 0
\]

So, the solutions to the equation are:
\[
x = 5 \quad \text{or} \quad x = -3
\]

### Step 2: Analyze the inequality
Now, we need to determine where \(x^2 - 2x - 15 > 0\), i.e., when the expression is positive. The boundary points divide the number line into three intervals:
1. \(x < -3\)
2. \(-3 < x < 5\)
3. \(x > 5\)

We will test each interval.

#### 1. For \(x < -3\) (e.g., \(x = -4\)):
\[
(x - 5)(x + 3) = (-4 - 5)(-4 + 3) = (-9)(-1) = 9 > 0
\]
The expression is positive.

#### 2. For \(-3 < x < 5\) (e.g., \(x = 0\)):
\[
(x - 5)(x + 3) = (0 - 5)(0 + 3) = (-5)(3) = -15 < 0
\]
The expression is negative.

#### 3. For \(x > 5\) (e.g., \(x = 6\)):
\[
(x - 5)(x + 3) = (6 - 5)(6 + 3) = (1)(9) = 9 > 0
\]
The expression is positive.

### Step 3: Conclusion
The solution to the inequality \(x^2 - 2x - 15 > 0\) is where the expression is positive:
\[
x \in (-\infty, -3) \cup (5, \infty)
\]

This is the solution to the inequality.

Do you want further details or have any questions?

Here are some related questions:
1. How would you solve \(x^2 - 2x - 15 \geq 0\)?
2. Can this inequality be solved using the quadratic formula?
3. What is the graphical interpretation of the solution?
4. How do we handle inequalities when the quadratic can't be factored easily?
5. What if the inequality was \(x^2 - 2x - 15 < 0\)?

**Tip:** Always test points within intervals to verify whether an inequality holds true in that region.

Learn how to solve the quadratic inequality x^2 - 2x - 15 > 0 through factoring and interval analysis. This detailed explanation includes factoring techniques, identifying boundary points, and testing intervals. Ideal for students in Grades 9-11.

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