1. **State the problem:** Solve the quadratic equation $$3x^2 = -8x - 2$$. 2. **Rewrite the equation in standard form:** Move all terms to one side: $$3x^2 + 8x + 2 = 0$$ 3. **Use the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=3$$, $$b=8$$, and $$c=2$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 8^2 - 4 \times 3 \times 2 = 64 - 24 = 40$$ 5. **Find the roots:** $$x = \frac{-8 \pm \sqrt{40}}{2 \times 3} = \frac{-8 \pm \sqrt{4 \times 10}}{6} = \frac{-8 \pm 2\sqrt{10}}{6}$$ 6. **Simplify the fraction:** $$x = \frac{\cancel{-8}}{\cancel{6}} \pm \frac{2\sqrt{10}}{6} = \frac{-4}{3} \pm \frac{\sqrt{10}}{3}$$ 7. **Final answer:** $$x = \frac{-4 + \sqrt{10}}{3} \quad \text{or} \quad x = \frac{-4 - \sqrt{10}}{3}$$