1. **State the problem:** Simplify the expression $$\frac{9x^2 - 3x - 2}{2x - 6} \cdot \frac{x - 3}{9x^2 - 4}$$. 2. **Factor all polynomials where possible:** - Factor numerator and denominator of the first fraction: $$9x^2 - 3x - 2 = (3x + 1)(3x - 2)$$ $$2x - 6 = 2(x - 3)$$ - Factor denominator of the second fraction: $$9x^2 - 4 = (3x - 2)(3x + 2)$$ 3. **Rewrite the expression with factored forms:** $$\frac{(3x + 1)(3x - 2)}{2(x - 3)} \cdot \frac{x - 3}{(3x - 2)(3x + 2)}$$ 4. **Cancel common factors:** - Cancel $x - 3$ from numerator and denominator: $$\frac{(3x + 1)(3x - 2)}{2\cancel{(x - 3)}} \cdot \frac{\cancel{x - 3}}{(3x - 2)(3x + 2)}$$ - Cancel $3x - 2$ from numerator and denominator: $$\frac{(3x + 1)\cancel{(3x - 2)}}{2} \cdot \frac{1}{\cancel{(3x - 2)}(3x + 2)} = \frac{3x + 1}{2} \cdot \frac{1}{3x + 2}$$ 5. **Multiply the remaining factors:** $$\frac{3x + 1}{2} \cdot \frac{1}{3x + 2} = \frac{3x + 1}{2(3x + 2)}$$ 6. **Final simplified expression:** $$\boxed{\frac{3x + 1}{2(3x + 2)}}$$ This is the simplified form of the original expression after factoring and canceling common terms.