1. **State the problem:** Solve the quadratic equation $$3x^2 + 5x + 6 = 0$$ using the discriminant formula. 2. **Recall the quadratic formula and discriminant:** The quadratic formula to find roots of $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{D}}{2a}$$ where the discriminant $$D = b^2 - 4ac$$. 3. **Identify coefficients:** Here, $$a = 3$$, $$b = 5$$, and $$c = 6$$. 4. **Calculate the discriminant:** $$D = b^2 - 4ac = 5^2 - 4 \times 3 \times 6 = 25 - 72 = -47$$ 5. **Interpret the discriminant:** Since $$D < 0$$, the equation has no real roots but two complex roots. 6. **Find the roots using the quadratic formula:** $$x = \frac{-5 \pm \sqrt{-47}}{2 \times 3} = \frac{-5 \pm i\sqrt{47}}{6}$$ 7. **Final answer:** $$x = \frac{-5}{6} + \frac{i\sqrt{47}}{6} \quad \text{and} \quad x = \frac{-5}{6} - \frac{i\sqrt{47}}{6}$$