1. **State the problem:** Find the roots of the quadratic equation $x^2 + 2x + 5 = 0$. 2. **Formula used:** The roots of a quadratic equation $ax^2 + bx + c = 0$ are given by the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** Here, $a = 1$, $b = 2$, and $c = 5$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times 5 = 4 - 20 = -16$$ 5. **Interpret the discriminant:** Since $\Delta < 0$, the roots are complex (no real roots). 6. **Calculate the roots:** $$x = \frac{-2 \pm \sqrt{-16}}{2 \times 1} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$ 7. **Final answer:** The roots are $x = -1 + 2i$ and $x = -1 - 2i$.