1. **State the problem:** Factor the polynomial $$2x^3 - 6x^2 - 4x + 12$$ by grouping. 2. **Step 1: Group the terms into two halves:** $$ (2x^3 - 6x^2) + (-4x + 12) $$ 3. **Step 2: Find the GCF of the first half:** The terms are $$2x^3$$ and $$-6x^2$$. The GCF is $$2x^2$$. Pulling it out: $$2x^2(x - 3)$$ 4. **Step 3: Find the GCF of the second half:** The terms are $$-4x$$ and $$12$$. The GCF is $$-4$$ (note the negative to match the sign inside the parentheses). Pulling it out: $$-4(x - 3)$$ 5. **Step 4: Write the expression with the factored groups:** $$2x^2(x - 3) - 4(x - 3)$$ 6. **Step 5: Factor out the common binomial factor $$(x - 3)$$:** $$\cancel{(x - 3)}(2x^2 - 4)\cancel{(x - 3)}$$ 7. **Step 6: Simplify the remaining expression:** $$2x^2 - 4$$ Factor out the GCF 2: $$2(x^2 - 2)$$ 8. **Step 7: Write the fully factored form:** $$ (x - 3) \times 2(x^2 - 2) = 2(x - 3)(x^2 - 2) $$ **Note:** The original problem's suggested factored form was $$(x - 3)(x^2 - 4)$$, but that is incorrect because the GCFs were not factored properly. **Final answer:** $$\boxed{2(x - 3)(x^2 - 2)}$$