1. We are asked to solve the quadratic equation $x^2 + 6x + 20 = 0$. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. Here, $a=1$, $b=6$, and $c=20$. 3. To solve, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = 6^2 - 4 \times 1 \times 20 = 36 - 80 = -44$$ 5. Since the discriminant is negative ($\Delta < 0$), there are no real solutions; the solutions are complex. 6. Compute the complex roots: $$x = \frac{-6 \pm \sqrt{-44}}{2} = \frac{-6 \pm \sqrt{44}i}{2} = \frac{-6 \pm 2\sqrt{11}i}{2} = -3 \pm \sqrt{11}i$$ 7. Therefore, the solutions are: $$x = -3 + \sqrt{11}i \quad \text{and} \quad x = -3 - \sqrt{11}i$$ These are the two complex roots of the quadratic equation.