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The expression given is:

\[
\frac{x^2 - 5x - 6}{x^2 + x - 6} \cdot \frac{x^2 - 2x - 15}{x^2 - 2x - 24}
\]

### Step 1: Factor Each Polynomial
Let's factor each of the polynomials:

1. \(x^2 - 5x - 6\) factors as \((x - 6)(x + 1)\)
2. \(x^2 + x - 6\) factors as \((x - 2)(x + 3)\)
3. \(x^2 - 2x - 15\) factors as \((x - 5)(x + 3)\)
4. \(x^2 - 2x - 24\) factors as \((x - 6)(x + 4)\)

### Step 2: Substitute the Factored Forms into the Expression
Substituting the factored forms back into the expression:

\[
\frac{(x - 6)(x + 1)}{(x - 2)(x + 3)} \cdot \frac{(x - 5)(x + 3)}{(x - 6)(x + 4)}
\]

### Step 3: Simplify the Expression
Next, cancel out any common factors in the numerator and denominator:

- The factor \((x - 6)\) appears in both the numerator and denominator, so it cancels out.
- The factor \((x + 3)\) also appears in both the numerator and denominator, so it cancels out.

After canceling, we are left with:

\[
\frac{(x + 1)(x - 5)}{(x - 2)(x + 4)}
\]

### Final Answer:
\[
\frac{(x + 1)(x - 5)}{(x - 2)(x + 4)}
\]

Would you like any further details or have any questions?

Here are 5 related questions to expand on this topic:
1. What are the methods to factor quadratic expressions?
2. How can you identify common factors in complex rational expressions?
3. What are the steps to simplify a product of rational expressions?
4. How do you determine the excluded values for rational expressions?
5. What role does the distributive property play in simplifying algebraic expressions?

**Tip:** Always factor completely before canceling terms in a rational expression to ensure that all possible simplifications are made.

Learn how to simplify the rational algebraic expression (x^2 - 5x - 6) / (x^2 + x - 6) * (x^2 - 2x - 15) / (x^2 - 2x - 24) step-by-step. This problem involves factoring polynomials and canceling common factors. Suitable for students in Grades 10-12.

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