1. **State the problem:** Solve the quadratic equation $x^2 + 2x - 5 = 0$ using the discriminant formula $D = b^2 - 4ac$. 2. **Identify coefficients:** Here, $a = 1$, $b = 2$, and $c = -5$. 3. **Calculate the discriminant:** $$D = b^2 - 4ac = 2^2 - 4 \times 1 \times (-5) = 4 + 20 = 24$$ 4. **Interpret the discriminant:** Since $D > 0$, there are two distinct real roots. 5. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{24}}{2 \times 1}$$ 6. **Simplify the square root:** $$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ 7. **Substitute back:** $$x = \frac{-2 \pm 2\sqrt{6}}{2}$$ 8. **Cancel common factor 2:** $$x = \frac{\cancel{2}(-1 \pm \sqrt{6})}{\cancel{2}} = -1 \pm \sqrt{6}$$ 9. **Final solutions:** $$x_1 = -1 + \sqrt{6}, \quad x_2 = -1 - \sqrt{6}$$ These are the two real roots of the quadratic equation.