1. State the problem: Solve the quadratic equation $$2x^2 + 8x + 12 = 0.$$\n\n2. Start by finding the discriminant $$\Delta$$ which tells us the nature of the roots: $$\Delta = b^2 - 4ac = 8^2 - 4 \times 2 \times 12 = 64 - 96 = -32.$$\n\n3. Since the discriminant is negative ($$-32 < 0$$), the equation has no real solutions, but two complex solutions.\n\n4. Use the quadratic formula to find the roots: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-8 \pm \sqrt{-32}}{2 \times 2} = \frac{-8 \pm \sqrt{-1 \times 32}}{4} = \frac{-8 \pm \sqrt{-1} \sqrt{32}}{4}.$$\n\n5. Recall $$\sqrt{-1} = i$$ (the imaginary unit), and simplify $$\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}.$$\n\n6. Substitute back in: $$x = \frac{-8 \pm 4i \sqrt{2}}{4} = \frac{-8}{4} \pm \frac{4i \sqrt{2}}{4} = -2 \pm i \sqrt{2}.$$\n\nFinal answer: The solutions are $$x = -2 + i \sqrt{2}$$ and $$x = -2 - i \sqrt{2}.$$