1. **State the problem:** We are given the quadratic equation $x^2 + 2x + 7 = 0$. 2. **Find the discriminant:** The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by the formula: $$\Delta = b^2 - 4ac$$ where $a=1$, $b=2$, and $c=7$. 3. **Calculate the discriminant:** $$\Delta = (2)^2 - 4 \times 1 \times 7 = 4 - 28 = -24$$ 4. **Interpret the discriminant:** Since $\Delta < 0$, the quadratic equation has two complex (non-real) roots. 5. **Use the quadratic formula to find the roots:** $$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$ Substitute the values: $$x = \frac{-2 \pm \sqrt{-24}}{2 \times 1}$$ 6. **Simplify the square root of the negative discriminant:** $$\sqrt{-24} = \sqrt{-1 \times 24} = i \sqrt{24} = i \sqrt{4 \times 6} = 2i \sqrt{6}$$ 7. **Write the exact solutions:** $$x = \frac{-2 \pm 2i \sqrt{6}}{2}$$ 8. **Simplify the fraction by canceling 2:** $$x = \frac{\cancel{2}(-1 \pm i \sqrt{6})}{\cancel{2}} = -1 \pm i \sqrt{6}$$ **Final answer:** The discriminant is $-24$, indicating two complex conjugate roots. The exact solutions are: $$x = -1 + i \sqrt{6} \quad \text{and} \quad x = -1 - i \sqrt{6}$$