1. **State the problem:** Solve the quadratic equation $$x^2 + 5x + 12 = 0$$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 1$$, $$b = 5$$, and $$c = 12$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 1 \times 12 = 25 - 48 = -23$$. 5. Since $$\Delta < 0$$, the equation has no real roots but two complex roots. 6. **Apply the quadratic formula:** $$x = \frac{-5 \pm \sqrt{-23}}{2 \times 1} = \frac{-5 \pm i\sqrt{23}}{2}$$. 7. **Final answer:** $$x = \frac{-5}{2} + \frac{i\sqrt{23}}{2} \quad \text{or} \quad x = \frac{-5}{2} - \frac{i\sqrt{23}}{2}$$. These are the two complex solutions to the quadratic equation.