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 $a=1$, $b=6$, and $c=-5$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 6^2 - 4 \times 1 \times (-5) = 36 + 20 = 56$$ 4. **Apply the quadratic formula:** $$x = \frac{-6 \pm \sqrt{56}}{2 \times 1} = \frac{-6 \pm \sqrt{56}}{2}$$ 5. **Simplify the square root:** $$\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}$$ 6. **Substitute back:** $$x = \frac{-6 \pm 2\sqrt{14}}{2}$$ 7. **Cancel common factor 2:** $$x = \frac{\cancel{2}(-3 \pm \sqrt{14})}{\cancel{2}} = -3 \pm \sqrt{14}$$ 8. **Final answer:** $$x = -3 + \sqrt{14} \quad \text{or} \quad x = -3 - \sqrt{14}$$