1. **State the problem:** Simplify the expression $$\frac{\left(-2 m^{-5} n^{4}\right)^{-2}}{\left(3 m^{8} n^{-4}\right)^{-1}}.$$\n\n2. **Recall the rules:**\n- Power of a product: $$(ab)^c = a^c b^c.$$\n- Power of a power: $$(a^b)^c = a^{bc}.$$\n- Negative exponent: $$a^{-b} = \frac{1}{a^b}.$$\n- Division of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}.$$\n\n3. **Apply the power to each factor in numerator:**\n$$\left(-2 m^{-5} n^{4}\right)^{-2} = (-2)^{-2} (m^{-5})^{-2} (n^{4})^{-2}.$$\n\n4. **Simplify each term:**\n$$(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4},$$\n$$(m^{-5})^{-2} = m^{(-5) \times (-2)} = m^{10},$$\n$$(n^{4})^{-2} = n^{4 \times (-2)} = n^{-8}.$$\n\n5. **Combine numerator:**\n$$\frac{1}{4} m^{10} n^{-8} = \frac{m^{10}}{4 n^{8}}.$$\n\n6. **Apply the power to each factor in denominator:**\n$$\left(3 m^{8} n^{-4}\right)^{-1} = 3^{-1} (m^{8})^{-1} (n^{-4})^{-1}.$$\n\n7. **Simplify each term:**\n$$3^{-1} = \frac{1}{3},$$\n$$(m^{8})^{-1} = m^{-8},$$\n$$(n^{-4})^{-1} = n^{4}.$$\n\n8. **Combine denominator:**\n$$\frac{1}{3} m^{-8} n^{4} = \frac{n^{4}}{3 m^{8}}.$$\n\n9. **Rewrite the original expression as:**\n$$\frac{\frac{m^{10}}{4 n^{8}}}{\frac{n^{4}}{3 m^{8}}} = \frac{m^{10}}{4 n^{8}} \times \frac{3 m^{8}}{n^{4}}.$$\n\n10. **Multiply numerators and denominators:**\n$$\frac{m^{10} \times 3 m^{8}}{4 n^{8} \times n^{4}} = \frac{3 m^{10+8}}{4 n^{8+4}} = \frac{3 m^{18}}{4 n^{12}}.$$\n\n**Final answer:** $$\boxed{\frac{3 m^{18}}{4 n^{12}}}.$$