1. **State the problem:** Solve the quadratic equation $$x^2 + 6x - 2 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 6$$, and $$c = -2$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 6^2 - 4 \times 1 \times (-2) = 36 + 8 = 44$$. 5. **Apply the quadratic formula:** $$x = \frac{-6 \pm \sqrt{44}}{2 \times 1} = \frac{-6 \pm \sqrt{44}}{2}$$. 6. **Simplify the square root:** $$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$$. 7. **Substitute back:** $$x = \frac{-6 \pm 2\sqrt{11}}{2}$$. 8. **Cancel common factor 2:** $$x = \frac{\cancel{2}(-3 \pm \sqrt{11})}{\cancel{2}} = -3 \pm \sqrt{11}$$. 9. **Final solutions:** $$x_1 = -3 + \sqrt{11}$$ $$x_2 = -3 - \sqrt{11}$$. These are the two roots of the quadratic equation.