Let's carefully examine the given polynomial and combine the like terms to express it correctly in standard form.
The given polynomial is:
[tex]\[ 8mn^5 - 2m^6 + 5m^2n^4 - m^3n^3 + n^6 - 4m^6 + 9m^2n^4 - mn^5 - 4m^3n^3 \][/tex]
Now, let's combine the like terms:
1. Combine terms with [tex]\(mn^5\)[/tex]:
[tex]\[ 8mn^5 - mn^5 = 7mn^5 \][/tex]
2. Combine terms with [tex]\(m^2n^4\)[/tex]:
[tex]\[ 5m^2n^4 + 9m^2n^4 = 14m^2n^4 \][/tex]
3. Combine terms with [tex]\(m^3n^3\)[/tex]:
[tex]\[ -m^3n^3 - 4m^3n^3 = -5m^3n^3 \][/tex]
4. Combine terms with [tex]\(m^6\)[/tex]:
[tex]\[ -2m^6 - 4m^6 = -6m^6 \][/tex]
5. The term [tex]\(n^6\)[/tex] remains as it is:
[tex]\[ n^6 \][/tex]
Bringing all these together, the polynomial in standard form is:
[tex]\[ n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6 \][/tex]
Therefore, the polynomial that correctly combines the like terms and expresses the given polynomial in standard form is:
[tex]\[ \boxed{n^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3 - 6m^6} \][/tex]
