1. **State the problem:** Simplify the expression $$(2x^2 - 2x - 12)(3x^2 - 10x + 3).$$ 2. **Factor each polynomial:** - For $2x^2 - 2x - 12$, factor out the common factor 2: $$2x^2 - 2x - 12 = 2(x^2 - x - 6).$$ - Factor $x^2 - x - 6$: $$x^2 - x - 6 = (x - 3)(x + 2).$$ So, $$2x^2 - 2x - 12 = 2(x - 3)(x + 2).$$ - For $3x^2 - 10x + 3$, find two numbers that multiply to $3 \times 3 = 9$ and add to $-10$; these are $-9$ and $-1$. Rewrite: $$3x^2 - 9x - x + 3 = 3x(x - 3) - 1(x - 3) = (3x - 1)(x - 3).$$ 3. **Rewrite the original expression using factors:** $$ (2(x - 3)(x + 2))((3x - 1)(x - 3)) = 2(3x - 1)(x - 3)(x - 3)(x + 2).$$ 4. **Simplify by combining like factors:** $$(x - 3)(x - 3) = (x - 3)^2,$$ so the expression becomes $$2(3x - 1)(x - 3)^2(x + 2).$$ **Final simplified form:** $$\boxed{2(3x - 1)(x - 3)^2(x + 2)}.$$