1. **State the problem:** Solve the quadratic equation $-x^2 - 6x - 4 = 0$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $-1$ to simplify the leading coefficient. $$-1 \times (-x^2 - 6x - 4) = -1 \times 0$$ $$\cancel{-1} \times (-x^2) + \cancel{-1} \times (-6x) + \cancel{-1} \times (-4) = 0$$ $$x^2 + 6x + 4 = 0$$ 3. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=6$, and $c=4$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 6^2 - 4 \times 1 \times 4 = 36 - 16 = 20$$ 5. **Find the roots:** $$x = \frac{-6 \pm \sqrt{20}}{2 \times 1} = \frac{-6 \pm \sqrt{20}}{2}$$ Simplify $\sqrt{20}$: $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ So, $$x = \frac{-6 \pm 2\sqrt{5}}{2}$$ 6. **Simplify the fraction:** $$x = \frac{\cancel{2}(-3 \pm \sqrt{5})}{\cancel{2}} = -3 \pm \sqrt{5}$$ 7. **Final answer:** $$x_1 = -3 + \sqrt{5}, \quad x_2 = -3 - \sqrt{5}$$