1. We are asked to solve the quadratic equation $x^2 + 12x - 8 = 0$ using the quadratic formula. 2. The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. For the equation $x^2 + 12x - 8 = 0$, we identify $a=1$, $b=12$, and $c=-8$. 4. Calculate the discriminant: $$\Delta = b^2 - 4ac = 12^2 - 4 \times 1 \times (-8) = 144 + 32 = 176$$ 5. Substitute into the quadratic formula: $$x = \frac{-12 \pm \sqrt{176}}{2 \times 1} = \frac{-12 \pm \sqrt{176}}{2}$$ 6. Simplify $\sqrt{176}$: $$\sqrt{176} = \sqrt{16 \times 11} = 4\sqrt{11}$$ 7. So, $$x = \frac{-12 \pm 4\sqrt{11}}{2}$$ 8. Simplify the fraction by dividing numerator and denominator by 2: $$x = \frac{\cancel{-12} \pm \cancel{4}\sqrt{11}}{\cancel{2}} = -6 \pm 2\sqrt{11}$$ 9. Final solutions: $$x_1 = -6 + 2\sqrt{11}$$ $$x_2 = -6 - 2\sqrt{11}$$