Divide. \[ \frac{3 x-6}{x^{2}+x-2} \div \frac{x-2}{x^{2}+5 x+6} \] Simplify your answer as much as possible. \( \square \)

To simplify the given expression, first, we rewrite the division as multiplication by the reciprocal: \[ \frac{3x - 6}{x^2 + x - 2} \div \frac{x - 2}{x^2 + 5x + 6} = \frac{3x - 6}{x^2 + x - 2} \times \frac{x^2 + 5x + 6}{x - 2}. \] Next, we factor where possible. 1. **Numerator \(3x - 6\)** can be factored as: \[ 3(x - 2). \] 2. **Denominator \(x^2 + x - 2\)** can be factored as: \[ (x - 1)(x + 2). \] 3. **Numerator \(x^2 + 5x + 6\)** can be factored as: \[ (x + 2)(x + 3). \] Now substituting these factored forms into our expression gives: \[ \frac{3(x - 2)}{(x - 1)(x + 2)} \times \frac{(x + 2)(x + 3)}{(x - 2)}. \] We can cancel \(x - 2\) in the numerator and denominator (assuming \(x \neq 2\)) and \(x + 2\): \[ = \frac{3}{x - 1} \times (x + 3). \] This simplifies to: \[ = \frac{3(x + 3)}{x - 1}. \] Finally, we can express the simplified result as: \[ \frac{3x + 9}{x - 1}. \] Thus, the final simplified result is: \[ \frac{3x + 9}{x - 1}. \]