1. **State the problem:** Solve the equation $$\frac{3 \sqrt{5y} + 1}{2} = -2$$ for $y$. 2. **Isolate the square root term:** Multiply both sides by 2 to eliminate the denominator. $$\cancel{2} \times \frac{3 \sqrt{5y} + 1}{\cancel{2}} = -2 \times 2$$ which simplifies to $$3 \sqrt{5y} + 1 = -4$$ 3. **Isolate the square root:** Subtract 1 from both sides. $$3 \sqrt{5y} + \cancel{1} - \cancel{1} = -4 - 1$$ $$3 \sqrt{5y} = -5$$ 4. **Divide both sides by 3:** $$\frac{3 \sqrt{5y}}{\cancel{3}} = \frac{-5}{\cancel{3}}$$ $$\sqrt{5y} = -\frac{5}{3}$$ 5. **Analyze the square root:** The square root function $\sqrt{x}$ always returns a non-negative value for real numbers. Since the right side is negative, there is no real solution. 6. **Conclusion:** There is no real value of $y$ that satisfies the equation. **Final answer:** No real solution.