20. \( \frac{m-2}{3 m+9} \cdot \frac{2 m+6}{2 m-4} \)

To simplify the expression: \[ \frac{m - 2}{3m + 9} \cdot \frac{2m + 6}{2m - 4} \] **Step 1: Factor where possible** 1. **Factor the denominators and numerators:** - \(3m + 9 = 3(m + 3)\) - \(2m + 6 = 2(m + 3)\) - \(2m - 4 = 2(m - 2)\) 2. **Rewrite the expression with these factors:** \[ \frac{m - 2}{3(m + 3)} \cdot \frac{2(m + 3)}{2(m - 2)} \] **Step 2: Cancel common factors** 1. **Cancel \(m - 2\) in the numerator and denominator:** \[ \frac{\cancel{m - 2}}{3(m + 3)} \cdot \frac{2(m + 3)}{2\cancel{(m - 2)}} \] 2. **Cancel \(m + 3\) in the numerator and denominator:** \[ \frac{1}{3\cancel{(m + 3)}} \cdot \frac{2\cancel{(m + 3)}}{2} \] 3. **Cancel the 2's:** \[ \frac{1}{3} \cdot \frac{\cancel{2}}{\cancel{2}} = \frac{1}{3} \] **Final Simplified Expression:** \[ \frac{1}{3} \] **Note:** - The expression is valid as long as the original denominators are not zero. Therefore, \(m \neq -3\) and \(m \neq 2\).