1. **State the problem:** Simplify the expression $$\left(\frac{3x^{2}y^{3}}{x^{-1}y^{2}}\right)^{-2}$$. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^{m} \div a^{n} = a^{m-n}$$. - When raising a power to another power, multiply exponents: $$(a^{m})^{n} = a^{mn}$$. 3. **Simplify inside the parentheses first:** $$\frac{3x^{2}y^{3}}{x^{-1}y^{2}} = 3 \times x^{2 - (-1)} \times y^{3 - 2} = 3x^{3}y^{1} = 3x^{3}y$$ 4. **Apply the outer exponent of -2:** $$\left(3x^{3}y\right)^{-2} = 3^{-2} \times x^{3 \times (-2)} \times y^{-2} = \frac{1}{3^{2}} \times x^{-6} \times y^{-2} = \frac{1}{9x^{6}y^{2}}$$ 5. **Final answer:** $$\boxed{\frac{1}{9x^{6}y^{2}}}$$