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The expression in the image appears to be a complex fraction with a division between two rational expressions. The expression is as follows:

\[
\frac{\frac{4x + 8}{x^2 - 4}}{\frac{x - 1}{x^2 + x - 6}}
\]

To simplify this, we can follow these steps:

1. **Factor the Denominators**:
   - For \( x^2 - 4 \), this is a difference of squares: \( x^2 - 4 = (x + 2)(x - 2) \).
   - For \( x^2 + x - 6 \), we factor it as \( (x + 3)(x - 2) \).

   So, rewriting the expression, we have:
   \[
   \frac{\frac{4x + 8}{(x + 2)(x - 2)}}{\frac{x - 1}{(x + 3)(x - 2)}}
   \]

2. **Simplify the Numerators**:
   - For \( 4x + 8 \), factor out 4: \( 4x + 8 = 4(x + 2) \).

   Now the expression becomes:
   \[
   \frac{\frac{4(x + 2)}{(x + 2)(x - 2)}}{\frac{x - 1}{(x + 3)(x - 2)}}
   \]

3. **Cancel Common Factors**:
   - In the numerator of the first fraction, \( (x + 2) \) appears in both the numerator and denominator, so we can cancel \( x + 2 \):
     \[
     \frac{\frac{4}{x - 2}}{\frac{x - 1}{(x + 3)(x - 2)}}
     \]

4. **Rewrite as Multiplication**:
   - To divide by a fraction, multiply by its reciprocal:
     \[
     \frac{4}{x - 2} \times \frac{(x + 3)(x - 2)}{x - 1}
     \]

5. **Cancel \( x - 2 \) Again**:
   - Now \( x - 2 \) cancels out in both the numerator and denominator:
     \[
     \frac{4(x + 3)}{x - 1}
     \]

So, the simplified form of the expression is:

\[
\frac{4(x + 3)}{x - 1}
\]

Would you like further details on any steps?

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Here are five related questions to expand your understanding:

1. How would the solution change if there were no common factors in the numerator and denominator?
2. What happens if we have a different factorization structure in one of the denominators?
3. Can you explain why factoring is essential in simplifying complex fractions?
4. What additional steps would be needed if there were additional terms in the numerators or denominators?
5. How does this process differ if we were dealing with addition instead of division?

**Tip:** Always start by factoring each term in the expression. This simplifies the problem and reveals common factors that can be canceled.

Simplify the complex fraction \( \frac{\frac{4x + 8}{x^2 - 4}}{\frac{x - 1}{x^2 + x - 6}} \).

Learn to simplify the complex fraction \( \frac{\frac{4x + 8}{x^2 - 4}}{\frac{x - 1}{x^2 + x - 6}} \) by factoring and canceling common terms. This solution involves understanding algebraic concepts such as factoring quadratics and simplifying complex fractions. Suitable for Grades 9-11.

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