1. **State the problem:** Solve the quadratic equation $x^2 + 6x - 5 = 0$ for $x$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = 1$, $b = 6$, and $c = -5$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 6^2 - 4 \times 1 \times (-5) = 36 + 20 = 56$$ 5. **Apply the quadratic formula:** $$x = \frac{-6 \pm \sqrt{56}}{2 \times 1} = \frac{-6 \pm \sqrt{56}}{2}$$ 6. **Simplify the square root:** $$\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}$$ 7. **Final solutions:** $$x = \frac{-6 \pm 2\sqrt{14}}{2} = -3 \pm \sqrt{14}$$ **Answer:** The solutions are $x = -3 + \sqrt{14}$ and $x = -3 - \sqrt{14}$.