he polynomial \( -2 m^{2} n^{3}+2 m^{3} n^{3}+7 n^{2}-6 m^{4} \) to be a binomial with a degree of 4 after it has been fully sin h must be the missing exponent on the \( m \) in the second term? 1 2 4

To determine the missing exponent on \( m \) in the second term of the polynomial \( -2 m^{2} n^{3} + 2 m^{x} n^{3} + 7 n^{2} - 6 m^{4} \) such that the polynomial becomes a binomial of degree 4, we first need to analyze the degrees of the terms. The degree of a term in a polynomial is the sum of the exponents of the variables in that term. Let's find the degrees of the existing terms: 1. The term \( -2 m^{2} n^{3} \) has a degree of \( 2 + 3 = 5 \). 2. The term \( 2 m^{x} n^{3} \) has a degree of \( x + 3 \). 3. The term \( 7 n^{2} \) has a degree of \( 2 \). 4. The term \( -6 m^{4} \) has a degree of \( 4 \). For the polynomial to be a binomial of degree 4, the highest degree term must be of degree 4. This means we need to ensure that the term with the highest degree is \( -6 m^{4} \) and that the other terms do not exceed this degree. To achieve this, we need to set the degree of the term \( 2 m^{x} n^{3} \) to be less than or equal to 4. Therefore, we have: \[ x + 3 \leq 4 \] Solving for \( x \): \[ x \leq 4 - 3 \] \[ x \leq 1 \] This means \( x \) can be 0 or 1. However, we also need to ensure that the term \( -2 m^{2} n^{3} \) does not dominate the polynomial. Since it has a degree of 5, it must be eliminated for the polynomial to be a binomial of degree 4. Thus, we can conclude that the only valid exponent for \( m \) in the second term that keeps the polynomial as a binomial of degree 4 is: \[ \boxed{1} \]