1. **State the problem:** Solve the quadratic equation $$x^2 + 2x - 8 = 0$$. 2. **Recall the formula:** For a quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=2$, and $c=-8$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times (-8) = 4 + 32 = 36$$ 4. **Find the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{36} = 6$$ 5. **Apply the quadratic formula:** $$x = \frac{-2 \pm 6}{2 \times 1} = \frac{-2 \pm 6}{2}$$ 6. **Calculate the two solutions:** - For the plus sign: $$x = \frac{-2 + 6}{2} = \frac{4}{2} = 2$$ - For the minus sign: $$x = \frac{-2 - 6}{2} = \frac{-8}{2} = -4$$ 7. **Final answer:** The solutions to the equation are $$x = 2$$ and $$x = -4$$.