1. **State the problem:** Simplify the expression $\left(2v - 3 \cdot u^{2}v - 12u^{3}v^{4}\right)^{-2}$. 2. **Rewrite the expression inside the parentheses:** Note that the expression is $2v - 3 \cdot u^{2}v - 12u^{3}v^{4}$. The multiplication is explicit only for the second term, so rewrite as $2v - 3u^{2}v - 12u^{3}v^{4}$. 3. **Factor the expression inside the parentheses:** Look for common factors in all terms. Each term contains a factor of $v$. So factor out $v$: $$2v - 3u^{2}v - 12u^{3}v^{4} = v(2 - 3u^{2} - 12u^{3}v^{3})$$ 4. **Rewrite the original expression using the factorization:** $$\left(v(2 - 3u^{2} - 12u^{3}v^{3})\right)^{-2}$$ 5. **Apply the power of a product rule:** $$\left(v\right)^{-2} \cdot \left(2 - 3u^{2} - 12u^{3}v^{3}\right)^{-2} = v^{-2} \cdot \left(2 - 3u^{2} - 12u^{3}v^{3}\right)^{-2}$$ 6. **Final simplified form:** $$\boxed{\frac{1}{v^{2} \left(2 - 3u^{2} - 12u^{3}v^{3}\right)^{2}}}$$ This is the simplified expression with the negative exponent removed and the common factor factored out.