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.