To correctly combine the like terms and express the given polynomial in standard form, we'll follow these steps:
1. Identify and group like terms:
- [tex]\( -2m^6 \)[/tex] and [tex]\( -4m^6 \)[/tex] are both like terms involving [tex]\( m^6 \)[/tex].
- [tex]\( 8mn^5 \)[/tex] and [tex]\( -mn^5 \)[/tex] are both like terms involving [tex]\( mn^5 \)[/tex].
- [tex]\( 5m^2n^4 \)[/tex] and [tex]\( 9m^2n^4 \)[/tex] are both like terms involving [tex]\( m^2n^4 \)[/tex].
- [tex]\( -m^3n^3 \)[/tex] and [tex]\( -4m^3n^3 \)[/tex] are both like terms involving [tex]\( m^3n^3 \)[/tex].
- [tex]\( n^6 \)[/tex] is a standalone term involving [tex]\( n^6 \)[/tex].
2. Sum the coefficients of each group of like terms:
- For [tex]\( m^6 \)[/tex] terms: [tex]\( -2 + -4 = -6 \)[/tex]
- For [tex]\( mn^5 \)[/tex] terms: [tex]\( 8 + -1 = 7 \)[/tex]
- For [tex]\( m^2n^4 \)[/tex] terms: [tex]\( 5 + 9 = 14 \)[/tex]
- For [tex]\( m^3n^3 \)[/tex] terms: [tex]\( -1 + -4 = -5 \)[/tex]
- The [tex]\( n^6 \)[/tex] term remains as [tex]\( 1 \)[/tex]
3. Write the polynomial combining these summed terms:
- The term for [tex]\( n^6 \)[/tex] is [tex]\( 1n^6 \)[/tex] or [tex]\( n^6 \)[/tex].
- The term for [tex]\( m^6 \)[/tex] is [tex]\( -6m^6 \)[/tex].
- The term for [tex]\( mn^5 \)[/tex] is [tex]\( 7mn^5 \)[/tex].
- The term for [tex]\( m^2n^4 \)[/tex] is [tex]\( 14m^2n^4 \)[/tex].
- The term for [tex]\( m^3n^3 \)[/tex] is [tex]\( -5m^3n^3 \)[/tex].
4. Combine all these terms to form the final polynomial:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
The polynomial that correctly combines like terms and is in standard form is:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
Hence, the correct polynomial is:
[tex]\[
n^6 - 6 m^6 + 7 m n^5 + 14 m^2 n^4 - 5 m^3 n^3
\][/tex]
which matches option four:
[tex]\[
n^6 - 6m^6 + 7mn^5 + 14m^2n^4 - 5m^3n^3
\][/tex]
