1. **State the problem:** Solve the equation $$x + 4 = x^2 + 8$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x + 4 = x^2 + 8 \implies 0 = x^2 + 8 - x - 4$$ 3. **Simplify the right side:** $$0 = x^2 - x + 4$$ 4. **Rewrite the quadratic equation:** $$x^2 - x + 4 = 0$$ 5. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=4$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 1 \times 4 = 1 - 16 = -15$$ 7. **Interpret the discriminant:** Since $\Delta < 0$, there are no real solutions; solutions are complex. 8. **Find the complex solutions:** $$x = \frac{-(-1) \pm \sqrt{-15}}{2 \times 1} = \frac{1 \pm \sqrt{-15}}{2} = \frac{1 \pm i\sqrt{15}}{2}$$ **Final answer:** $$x = \frac{1}{2} + \frac{i\sqrt{15}}{2} \quad \text{or} \quad x = \frac{1}{2} - \frac{i\sqrt{15}}{2}$$