https://file.aiquickdraw.com/mathbot/file/1729622053919k2zt6xpq.jpg

The expression you uploaded is:

\[
\frac{\frac{4}{x+1} - 2}{3 - \frac{9}{x+2}}
\]

### Steps to simplify:

#### 1. **Simplify the numerator:**
The numerator is:
\[
\frac{4}{x+1} - 2
\]

To subtract, we need a common denominator:
\[
\frac{4}{x+1} - 2 = \frac{4}{x+1} - \frac{2(x+1)}{x+1} = \frac{4 - 2(x+1)}{x+1} = \frac{4 - 2x - 2}{x+1} = \frac{2 - 2x}{x+1}
\]

#### 2. **Simplify the denominator:**
The denominator is:
\[
3 - \frac{9}{x+2}
\]

Similarly, we get a common denominator:
\[
3 - \frac{9}{x+2} = \frac{3(x+2)}{x+2} - \frac{9}{x+2} = \frac{3x + 6 - 9}{x+2} = \frac{3x - 3}{x+2}
\]

#### 3. **Write the full expression:**
Now, the expression is:
\[
\frac{\frac{2 - 2x}{x+1}}{\frac{3x - 3}{x+2}}
\]

This can be written as:
\[
\frac{2 - 2x}{x+1} \times \frac{x+2}{3x - 3}
\]

#### 4. **Factor and simplify:**
We can factor out common terms:
\[
\frac{2 - 2x}{x+1} = \frac{-2(x-1)}{x+1}
\]
\[
\frac{3x - 3}{x+2} = 3(x-1)
\]

Thus, the expression becomes:
\[
\frac{-2(x-1)}{x+1} \times \frac{x+2}{3(x-1)} = \frac{-2}{3(x+1)}
\]

### Final Simplified Expression:
\[
\boxed{\frac{-2}{3(x+1)}}
\]

Would you like further clarification or additional details on this solution?

### Related Questions:
1. How do you find common denominators for rational expressions?
2. What are the rules for factoring polynomials?
3. How can you simplify complex fractions involving variables?
4. What are other methods to simplify rational expressions?
5. How do you solve rational equations?

### Tip:
Always check for common factors in both the numerator and denominator when simplifying complex rational expressions. This can greatly reduce the complexity of the problem.

Simplify the expression \( \frac{\frac{4}{x+1} - 2}{3 - \frac{9}{x+2}} \).

Learn how to simplify the complex rational expression \( \frac{\frac{4}{x+1} - 2}{3 - \frac{9}{x+2}} \) using factorization and common denominator methods. This guide walks you through each step, focusing on algebraic simplification techniques. Suitable for students in Grades 9-11.

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