We are asked to multiply and simplify the following rational expressions:

$\frac{(x-12)(x+2)}{x+5} \times \frac{x+5}{(x+7)(x-4)}$

Step-by-Step Solution

1. Multiply the two fractions: To multiply two fractions, we multiply the numerators and multiply the denominators:

   $\frac{(x-12)(x+2)}{x+5} \times \frac{x+5}{(x+7)(x-4)} = \frac{(x-12)(x+2)(x+5)}{(x+5)(x+7)(x-4)}$

2. Cancel out common factors: The factor $(x+5)$ appears in both the numerator and the denominator, so we can cancel it out:

   $\frac{(x-12)(x+2)}{(x+7)(x-4)}$

3. Final simplified expression: The simplified expression is:

   $\frac{(x-12)(x+2)}{(x+7)(x-4)}$

Conclusion:

The simplified form of the given rational expression is:

$\frac{(x-12)(x+2)}{(x+7)(x-4)}$

Would you like any further clarification on this, or perhaps more details? Here are 5 follow-up questions to deepen your understanding:

1. How would the solution change if one of the terms was negative?
2. Can you factor the numerator $(x-12)(x+2)$ and the denominator $(x+7)(x-4)$ further?
3. How do you handle rational expressions when there are no common factors?
4. What happens if $x$ takes values that make any denominator equal to zero?
5. How would this process differ if we were asked to divide instead of multiply?

Tip: When simplifying rational expressions, always check for common factors that can be canceled before expanding the expressions.