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The given mathematical expression is:

\[
\frac{x^2 - 5x + 6}{x^2 - x - 6} \div \frac{x^2 + x - 6}{x^2 + 5x + 6}
\]

### Step 1: Factor both numerators and denominators

1. Factor the quadratic expressions:

- \( x^2 - 5x + 6 \) factors as \( (x - 2)(x - 3) \).
- \( x^2 - x - 6 \) factors as \( (x - 3)(x + 2) \).
- \( x^2 + x - 6 \) factors as \( (x - 2)(x + 3) \).
- \( x^2 + 5x + 6 \) factors as \( (x + 2)(x + 3) \).

### Step 2: Rewrite the expression

After factoring, the expression becomes:

\[
\frac{(x - 2)(x - 3)}{(x - 3)(x + 2)} \div \frac{(x - 2)(x + 3)}{(x + 2)(x + 3)}
\]

### Step 3: Convert the division into multiplication

When dividing fractions, we multiply by the reciprocal. The expression becomes:

\[
\frac{(x - 2)(x - 3)}{(x - 3)(x + 2)} \times \frac{(x + 2)(x + 3)}{(x - 2)(x + 3)}
\]

### Step 4: Simplify the expression

Cancel out the common factors in the numerator and denominator:

- \( (x - 3) \) and \( (x + 2) \) cancel out.
- \( (x - 2) \) and \( (x + 3) \) cancel out.

After canceling, the expression simplifies to:

\[
1
\]

Thus, the final simplified result is:

\[
1
\]

Would you like more details on any of the steps? Here are some related questions to explore further:

1. How do you factor quadratic expressions like \(x^2 - 5x + 6\)?
2. What is the process for converting division of fractions into multiplication?
3. Why does canceling common factors simplify the expression?
4. What happens if you can't factor certain quadratic expressions?
5. How does this process work for more complex rational expressions?

**Tip**: Always check if quadratic expressions can be factored before simplifying rational expressions. It helps in canceling common terms easily.

Simplify the expression: \( \frac{x^2 - 5x + 6}{x^2 - x - 6} \div \frac{x^2 + x - 6}{x^2 + 5x + 6} \).

Learn how to simplify rational expressions by factoring quadratic expressions and using division of fractions. The problem involves factoring, simplifying, and canceling common terms to find the final result. Suitable for students in Grades 9-10.

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