1. **State the problem:** Simplify the expression $$\frac{2x^2 - 2x - 12}{3x^2 - 10x + 3}$$ and determine which of the given options matches the simplified form. 2. **Factor numerator and denominator:** - Numerator: $$2x^2 - 2x - 12 = 2(x^2 - x - 6)$$ - Factor inside parentheses: $$x^2 - x - 6 = (x - 3)(x + 2)$$ - So numerator becomes $$2(x - 3)(x + 2)$$ - Denominator: $$3x^2 - 10x + 3$$ - Find factors of $$3 \times 3 = 9$$ that sum to $$-10$$: $$-9$$ and $$-1$$ - Rewrite: $$3x^2 - 9x - x + 3$$ - Factor by grouping: $$3x(x - 3) - 1(x - 3) = (3x - 1)(x - 3)$$ 3. **Rewrite the fraction:** $$\frac{2(x - 3)(x + 2)}{(3x - 1)(x - 3)}$$ 4. **Cancel common factors:** $$\frac{2\cancel{(x - 3)}(x + 2)}{(3x - 1)\cancel{(x - 3)}} = \frac{2(x + 2)}{3x - 1}$$ 5. **Simplify numerator:** $$2(x + 2) = 2x + 4$$ 6. **Final simplified expression:** $$\frac{2x + 4}{3x - 1}$$ 7. **Compare with options:** - Option c) $$\frac{2x + 4}{3x - 1}$$ matches the simplified expression. **Answer:** Option c) $$\frac{2x + 4}{3x - 1}$$