1. **State the problem:** Solve the quadratic equation $x^2 + 6x + 4 = 0$ 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 $D = b^2 - 4ac$ determines the nature of the roots. 3. **Identify coefficients:** Here, $a = 1$, $b = 6$, and $c = 4$. 4. **Calculate the discriminant:** $$D = b^2 - 4ac = 6^2 - 4 \times 1 \times 4 = 36 - 16 = 20$$ 5. **Evaluate the roots using the quadratic formula:** $$x = \frac{-6 \pm \sqrt{20}}{2 \times 1} = \frac{-6 \pm \sqrt{20}}{2}$$ 6. **Simplify the square root:** $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ 7. **Substitute back:** $$x = \frac{-6 \pm 2\sqrt{5}}{2}$$ 8. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2}(-3 \pm \sqrt{5})}{\cancel{2}} = -3 \pm \sqrt{5}$$ 9. **Final answer:** $$x_1 = -3 + \sqrt{5}, \quad x_2 = -3 - \sqrt{5}$$