What is the quotient \( \frac{2 m^{9} n^{4}}{-4 m^{-3} n^{-2}} \) in simplest form? Assume \( m \neq 0, n \neq 0 \). - \( \begin{array}{l}m^{12} n^{6} \\ 2 \\ -\frac{m^{27} n^{8}}{2} \\ 6 m^{12} n^{6} \\ 8 m^{12} n^{6}\end{array} \)

To simplify the quotient \( \frac{2 m^{9} n^{4}}{-4 m^{-3} n^{-2}} \), we first simplify the coefficients: \[ \frac{2}{-4} = -\frac{1}{2} \] Next, we'll simplify the \( m \) terms. Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we have: \[ \frac{m^{9}}{m^{-3}} = m^{9 - (-3)} = m^{9 + 3} = m^{12} \] Similarly, for the \( n \) terms: \[ \frac{n^{4}}{n^{-2}} = n^{4 - (-2)} = n^{4 + 2} = n^{6} \] Now, bringing it all together, we have: \[ \frac{2 m^{9} n^{4}}{-4 m^{-3} n^{-2}} = -\frac{1}{2} m^{12} n^{6} \] So the simplest form of the expression is: \[ -\frac{1}{2} m^{12} n^{6} \] However, since we need to choose from the provided options, we'll acknowledge that neither matches this form directly. But based on our simplification, we can conclude that the expression leads to a form where the main variable components match \( m^{12} n^{6} \) factors. Among the given options, none of them directly produce a negative coefficient as we derived, but if we need to choose an equivalent non-negative form without losing the essence of \( m^{12} n^{6} \), we could assume an assessment based on purely the variable aspects. From there, the result in terms of structure signals that the multiple choice answers primarily suggest combinations which are consistent with \( m^{12} n^{6} \) leading us to decide towards potential multiples like \( 6 m^{12} n^{6} \) and \( 8 m^{12} n^{6} \) in consideration of overall context, where simply the signs were different than presented in the answer. In summary, none of the options align perfectly with the calculated output, yet recognizing \( m^{12} n^{6} \) is a core component amidst fracturing metrics of 2 potentially guides deductions towards problem setups.