1. **State the problem:** Solve the equation $$\sqrt{x^2 + 3x + 20} = 0$$. 2. **Recall the property of square roots:** The square root of a number is zero if and only if the number inside the root is zero. That is, $$\sqrt{A} = 0 \implies A = 0$$, where $$A \geq 0$$. 3. **Apply this property:** Set the expression inside the square root equal to zero: $$x^2 + 3x + 20 = 0$$ 4. **Analyze the quadratic equation:** The quadratic is $$x^2 + 3x + 20 = 0$$. 5. **Calculate the discriminant to check for real roots:** $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times 20 = 9 - 80 = -71$$ 6. **Interpret the discriminant:** Since $$\Delta < 0$$, there are no real solutions to the quadratic equation. 7. **Conclusion:** Because the expression inside the square root never equals zero for any real $$x$$, the original equation has **no real solutions**. **Final answer:** No real solutions exist for $$\sqrt{x^2 + 3x + 20} = 0$$.