Let's examine the given polynomial and simplify it step-by-step:
The polynomial provided is:
[tex]\[ -2 m^2 n^3 + 2 m^x n^y + 7 n^2 - 6 m^4 \][/tex]
We are given that this polynomial needs to be simplified, and we want it to become a binomial of degree 4.
The polynomial after simplification should look like this:
[tex]\[ -2 m^2 n^3 + 2 m^x n^y + 7 n^2 - 6 m^4 \][/tex]
First, we observe that in order to get a binomial of degree 4 after simplification, the terms need to combine to leave us with two terms, and these must both be of degree 4.
Let's simplify the polynomial by first grouping like terms:
We see that [tex]\(-2 m^2 n^3\)[/tex] and [tex]\(2 m^x n^y\)[/tex] are both terms involving [tex]\(m\)[/tex] and [tex]\(n\)[/tex]. Let us combine these terms by selecting exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] such that these two terms cancel each other out.
To understand this, we know:
[tex]\[ -2 m^2 n^3 + 2 m^x n^y \][/tex]
For these terms to cancel out:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 \][/tex]
We observe that:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 3 \][/tex]
Thus, the term [tex]\(2 m^x n^y\)[/tex] should be:
[tex]\[ 2 m^2 n^3\][/tex]
Now we see that:
[tex]\[ -2 m^2 n^3 + 2 m^2 n^3 + 7 n^2 - 6 m^4 \][/tex]
The [tex]\(-2 m^2 n^3\)[/tex] and [tex]\(2 m^2 n^3\)[/tex] terms cancel out fully and leave us with:
[tex]\[ 7 n^2 - 6 m^4 \][/tex]
Lastly let's consider the degrees of all remaining terms:
7 [tex]\(n^2\)[/tex] has a degree of 2 as [tex]\(n^2\)[/tex]
-6 [tex]\(m^4\)[/tex] has a degree of 4 as [tex]\(m^4\)[/tex]
Thus, the simplified polynomial is:
[tex]\[ 7 n^2 - 6 m^4\][/tex]
This polynomial is a binomial, containing two terms. We needed this polynomial to be of degree 4.
Therefore, the missing exponent on [tex]\( m \)[/tex] in the second term is:
[tex]\[ \boxed{2} \][/tex]
