To determine the missing exponent on the [tex]\( m \)[/tex] in the second term so that the polynomial [tex]\( -2 m^2 n^3 + 2 m^x n^3 + 7 n^2 - 6 m^4 \)[/tex] can be simplified into a binomial with a degree of 4, let's go through the following steps:
1. Identify and Combine Like Terms:
- The given polynomial is [tex]\( -2 m^2 n^3 + 2 m^x n^3 + 7 n^2 - 6 m^4 \)[/tex].
- Combine like terms involving the same powers of [tex]\( m \)[/tex] and [tex]\( n \)[/tex].
2. Combine the provided terms:
- Consider the terms [tex]\( -2 m^2 n^3 \)[/tex] and [tex]\( 2 m^x n^3 \)[/tex].
- For these terms to combine and cancel out each other, the exponents on [tex]\( m \)[/tex] must be the same. This will mean [tex]\( x = 2 \)[/tex].
3. Simplify the Polynomial:
- By setting [tex]\( x = 2 \)[/tex], the terms are [tex]\( -2 m^2 n^3 + 2 m^2 n^3 \)[/tex]. These terms cancel out each other because they are additive inverses.
- After canceling these terms, we are left with [tex]\( 7 n^2 - 6 m^4 \)[/tex].
4. Identify the Resulting Polynomial Degree and Binomial Condition:
- The polynomial [tex]\( 7 n^2 - 6 m^4 \)[/tex] now has two terms: [tex]\( 7 n^2 \)[/tex] and [tex]\( -6 m^4 \)[/tex].
- For the polynomial to be a binomial with a degree of 4, we need to ensure that the highest degree term is 4.
- The term [tex]\( 7 n^2 \)[/tex] has a degree of 2, while the term [tex]\( -6 m^4 \)[/tex] has a degree of 4.
5. Conclusion:
- To meet the condition of being a binomial of degree 4, the only significant term should be [tex]\( -6 m^4 \)[/tex] which already has a degree of 4.
- Thus, we find that setting [tex]\( x = 2 \)[/tex] achieves this, turning the polynomial [tex]\( -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \)[/tex] into a simplified binomial with the leading term [tex]\( -6 m^4 \)[/tex].
Hence, the missing exponent on the [tex]\( m \)[/tex] in the second term must be [tex]\( \boxed{2} \)[/tex].
