https://file.aiquickdraw.com/mathbot/file/17258380562916fm439mj.jpg

The expression shown in the image is:

\[
\frac{(x - 9)(x + 7)}{x + 1} \cdot \frac{x}{9 - x}
\]

### Step-by-step simplification:

1. **Factorize and rewrite**: Notice that \( 9 - x \) can be rewritten as \( -(x - 9) \), so we can simplify:
   \[
   \frac{(x - 9)(x + 7)}{x + 1} \cdot \frac{x}{-(x - 9)}
   \]

2. **Cancel common terms**: The term \( x - 9 \) appears in both the numerator and denominator, so we can cancel it out:
   \[
   \frac{(x + 7)}{x + 1} \cdot \frac{-x}{1}
   \]

3. **Multiply the remaining terms**: Multiply the numerators and denominators:
   \[
   \frac{-(x)(x + 7)}{x + 1}
   \]

4. **Final expression**: Distribute the \( x \) in the numerator:
   \[
   \frac{-x^2 - 7x}{x + 1}
   \]

This is the simplified form of the given expression.

Let me know if you would like further details or have any questions.

### Relative Questions:
1. What is the importance of recognizing when to factor terms in expressions?
2. Can you explain why \( 9 - x \) is rewritten as \( -(x - 9) \)?
3. How does multiplying fractions work in algebra?
4. What happens if you substitute specific values for \( x \) in the simplified expression?
5. Are there any values of \( x \) that make the original or simplified expression undefined?

### Tip:
Always check for common factors before multiplying or expanding expressions—canceling common terms can greatly simplify the problem.

Simplify the following expression: (x - 9)(x + 7) / (x + 1) * x / (9 - x)

This problem involves simplifying a rational algebraic expression by factoring, canceling common terms, and recognizing when to rewrite terms like 9 - x as -(x - 9). A step-by-step solution is provided for students in Grade 9-10.

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