1. **State the problem:** Simplify and solve the differential equation $$3(y + 2) \, dx - x y \, dy = 0$$. 2. **Rewrite the equation:** We have $$3(y + 2) \, dx = x y \, dy$$ or equivalently $$\frac{dx}{dy} = \frac{x y}{3(y + 2)}$$. 3. **Separate variables:** Rewrite as $$\frac{1}{x} \, dx = \frac{y}{3(y + 2)} \, dy$$. 4. **Integrate both sides:** Left side: $$\int \frac{1}{x} \, dx = \ln|x| + C_1$$ Right side: $$\int \frac{y}{3(y + 2)} \, dy = \frac{1}{3} \int \frac{y}{y + 2} \, dy$$. 5. **Simplify the integrand:** $$\frac{y}{y + 2} = 1 - \frac{2}{y + 2}$$. 6. **Integrate the right side:** $$\frac{1}{3} \int \left(1 - \frac{2}{y + 2}\right) dy = \frac{1}{3} \left( y - 2 \ln|y + 2| \right) + C_2$$. 7. **Combine results:** $$\ln|x| = \frac{y}{3} - \frac{2}{3} \ln|y + 2| + C$$ where $$C = C_2 - C_1$$. 8. **Multiply both sides by 3:** $$3 \ln|x| = y - 2 \ln|y + 2| + C'$$. 9. **Final implicit solution:** $$3 \ln|x| = y - 2 \ln|y + 2| + C'$$. This matches option **b**. **Answer:** b. $$3 \ln|x| = y - 2 \ln|y + 2| + C$$