1. **State the problem:** Factor the expression $$3x^2 + 6x^2 - 4x - 8$$. 2. **Combine like terms:** First, combine the terms with $$x^2$$. $$3x^2 + 6x^2 = 9x^2$$ So the expression becomes: $$9x^2 - 4x - 8$$ 3. **Use factoring by grouping:** The expression has three terms, but we can treat it as a quadratic in standard form: $$9x^2 - 4x - 8$$ 4. **Find two numbers that multiply to $$a \times c = 9 \times (-8) = -72$$ and add to $$b = -4$$. These numbers are $$-12$$ and $$6$$ because $$-12 \times 6 = -72$$ and $$-12 + 6 = -6$$ (which is not $$-4$$). Since this doesn't work, try another approach. 5. **Try factoring out the greatest common factor (GCF):** The GCF of all terms is 1, so no common factor. 6. **Use the quadratic formula to factor:** The quadratic is $$9x^2 - 4x - 8$$. The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=9$$, $$b=-4$$, $$c=-8$$. Calculate the discriminant: $$b^2 - 4ac = (-4)^2 - 4 \times 9 \times (-8) = 16 + 288 = 304$$ Calculate the roots: $$x = \frac{4 \pm \sqrt{304}}{18} = \frac{4 \pm 4\sqrt{19}}{18} = \frac{2 \pm 2\sqrt{19}}{9}$$ 7. **Write the factored form using roots:** $$9x^2 - 4x - 8 = 9 \left(x - \frac{2 + 2\sqrt{19}}{9}\right) \left(x - \frac{2 - 2\sqrt{19}}{9}\right)$$ 8. **Simplify the factors:** $$= (9x - 2 - 2\sqrt{19})(9x - 2 + 2\sqrt{19})$$ **Final answer:** $$3x^2 + 6x^2 - 4x - 8 = (9x - 2 - 2\sqrt{19})(9x - 2 + 2\sqrt{19})$$