https://file.aiquickdraw.com/mathbot/file/1724622413416x5dyai9e.jpg

The expression given in the image is:

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

To solve this, we need to simplify the complex fraction.

### Step 1: Factorize the denominators
1. **Factor \( x^2 - 1 \)**:  
   \( x^2 - 1 \) is a difference of squares, so it can be factorized as:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]

2. **Factor \( x^2 + 2x - 3 \)**:  
   We need to factorize this quadratic expression:
   \[
   x^2 + 2x - 3 = (x + 3)(x - 1)
   \]

### Step 2: Substitute the factorizations back into the expression
Now, substitute the factored forms into the expression:
\[
\frac{\frac{x+2}{(x-1)(x+1)}}{\frac{x-3}{(x+3)(x-1)}}
\]

### Step 3: Simplify the complex fraction
The complex fraction can be simplified by multiplying the numerator of the outer fraction by the reciprocal of the denominator of the outer fraction:
\[
\frac{x+2}{(x-1)(x+1)} \times \frac{(x+3)(x-1)}{x-3}
\]

### Step 4: Cancel out common factors
Notice that \( (x-1) \) is a common factor in both the numerator and denominator, so it can be canceled out:
\[
\frac{(x+2)(x+3)}{(x+1)(x-3)}
\]

This is the simplified expression.

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

Would you like to know more about how each step was performed or have any other questions? Here are some related questions you might find interesting:

1. How do you factorize a quadratic expression like \(x^2 + 2x - 3\)?
2. What is the difference of squares and how do you recognize it?
3. How can you simplify complex fractions in general?
4. What are some common mistakes when canceling out terms in fractions?
5. How do you handle expressions with more complex factorization challenges?

**Tip:** When simplifying fractions, always check for common factors in both the numerator and denominator before proceeding to multiply or divide. This can save time and reduce the complexity of the problem.

Learn how to simplify the complex fraction (x+2)/(x^2-1) / (x-3)/(x^2+2x-3) step-by-step with detailed explanations and examples. This problem covers key algebraic concepts, including fraction simplification and factoring quadratic expressions. Suitable for students in Grades 9-12.

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