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The question asks to simplify the following expression:

\[
\frac{x^2 + x - 2}{x^2 - 3x + 2} \div \frac{2x + 4}{x^2 - 4}
\]

To solve this, let's go through the steps:

### Step 1: Simplify Each Fraction

1. **First Fraction**: \(\frac{x^2 + x - 2}{x^2 - 3x + 2}\)

   - Factor the numerator: \(x^2 + x - 2 = (x + 2)(x - 1)\)
   - Factor the denominator: \(x^2 - 3x + 2 = (x - 1)(x - 2)\)

   So, we can rewrite this fraction as:
   \[
   \frac{(x + 2)(x - 1)}{(x - 1)(x - 2)}
   \]

   Cancel out \((x - 1)\) from the numerator and denominator:
   \[
   \frac{x + 2}{x - 2}
   \]

2. **Second Fraction**: \(\frac{2x + 4}{x^2 - 4}\)

   - Factor the numerator: \(2x + 4 = 2(x + 2)\)
   - Factor the denominator (difference of squares): \(x^2 - 4 = (x + 2)(x - 2)\)

   So, this fraction becomes:
   \[
   \frac{2(x + 2)}{(x + 2)(x - 2)}
   \]

   Cancel out \((x + 2)\) from the numerator and denominator:
   \[
   \frac{2}{x - 2}
   \]

### Step 2: Divide the Fractions

Now we have:
\[
\frac{x + 2}{x - 2} \div \frac{2}{x - 2}
\]

Dividing by a fraction is equivalent to multiplying by its reciprocal:
\[
\frac{x + 2}{x - 2} \times \frac{x - 2}{2}
\]

Cancel \((x - 2)\) from the numerator and denominator:
\[
\frac{x + 2}{2}
\]

### Final Answer

The simplified form of the expression is:
\[
\frac{x + 2}{2}
\]

Would you like a detailed explanation of each factoring step, or do you have any specific questions?

### Related Questions

1. How do you simplify expressions involving division of rational functions?
2. What are the methods for factoring quadratic expressions?
3. How do you handle division by a fraction in algebra?
4. What is the process for simplifying complex rational expressions?
5. How does factoring help in simplifying algebraic fractions?

### Tip
When simplifying rational expressions, always look for common factors in the numerator and denominator to cancel out terms easily.

Simplify \( \frac{x^2 + x - 2}{x^2 - 3x + 2} \div \frac{2x + 4}{x^2 - 4} \)

This page provides a step-by-step solution to simplify the rational expression (x^2 + x - 2) / (x^2 - 3x + 2) ÷ (2x + 4) / (x^2 - 4). Key algebraic concepts such as factoring, difference of squares, and simplification of rational expressions are covered. Suitable for high school students in Grades 10-12.

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