Multiply or divide the following rational expressions, as indicated, and simplify your answer.

(x-12)(x+2)/x+5 * x+5/(x+7)(x-4)

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.

Multiply or divide the following rational expressions, as indicated, and simplify your answer: (x-12)(x+2)/x+5 * x+5/(x+7)(x-4)

Learn how to simplify the rational expression (x-12)(x+2)/x+5 * x+5/(x+7)(x-4) by multiplying and canceling common factors. This step-by-step solution covers essential algebra concepts including factoring and rational expression simplification. Suitable for students in Grades 9-11.

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