1. **State the problem:** Solve the quadratic equation $$x^2 + 10x + 40 = 0$$ for $x$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solutions for $x$ are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients. 3. **Identify coefficients:** Here, $a = 1$, $b = 10$, and $c = 40$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 10^2 - 4 \times 1 \times 40 = 100 - 160 = -60$$ 5. **Interpret the discriminant:** Since $$\Delta < 0$$, the equation has no real solutions but two complex conjugate solutions. 6. **Find the solutions:** $$x = \frac{-10 \pm \sqrt{-60}}{2 \times 1} = \frac{-10 \pm \sqrt{60}i}{2} = \frac{-10 \pm \sqrt{4 \times 15}i}{2} = \frac{-10 \pm 2\sqrt{15}i}{2}$$ 7. **Simplify:** $$x = -5 \pm \sqrt{15}i$$ **Final answer:** $$x = -5 + \sqrt{15}i$$ or $$x = -5 - \sqrt{15}i$$