Let's carefully analyze the provided polynomial expression [tex]\( -2m^2 n^3 + 2m^2 n^3 + 7n^2 - 6m^4 \)[/tex] and simplify it step-by-step.
1. Combine like terms: Notice that [tex]\( -2m^2 n^3 \)[/tex] and [tex]\( 2m^2 n^3 \)[/tex] are like terms because they have the same variables raised to the same power.
[tex]\[
-2m^2 n^3 + 2m^2 n^3 = 0
\][/tex]
So, the polynomial simplifies to:
[tex]\[
7n^2 - 6m^4
\][/tex]
2. Identify the resulting polynomial: After simplification, the polynomial is [tex]\( 7n^2 - 6m^4 \)[/tex].
3. Determine the degree of the polynomial: The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in standard form.
[tex]\[
7n^2 \text{ has degree } 2
\][/tex]
[tex]\[
-6m^4 \text{ has degree } 4
\][/tex]
Therefore, the degree of the simplified polynomial [tex]\( 7n^2 - 6m^4 \)[/tex] is 4.
4. Convert the polynomial into a binomial: To convert the polynomial [tex]\( 7n^2 - 6m^4 \)[/tex] into a binomial form with a specified missing exponent on [tex]\( m \)[/tex] for the second term, we should take note of the form where we have a term involving [tex]\( n \)[/tex] and another involving [tex]\( m \)[/tex].
Given the polynomial results in a binomial, the necessary form already is a binomial: [tex]\( 7n^2 - 6m^4 \)[/tex].
Here, the term with [tex]\( n \)[/tex] is [tex]\( 7n^2 \)[/tex] and the term with [tex]\( m \)[/tex] is [tex]\( -6m^4 \)[/tex]. There is no need to hypothesize about a missing exponent in this context because the polynomial simplifies directly to a binomial with [tex]\( n \)[/tex] and [tex]\( m \)[/tex] terms distinctly shown.
5. Conclusion: The exponent on [tex]\( m \)[/tex] in the term [tex]\( -6m^4 \)[/tex] is already given as 4, which is the highest degree exponent resulting in this simplified binomial form.
Therefore, the missing exponent needed on [tex]\( m \)[/tex] in the context provided earlier is indeed the exponent [tex]\( 4 \)[/tex] as given in the term [tex]\( -6m^4 \)[/tex].
