Let's solve the polynomial subtraction step-by-step:
1. Write subtraction as addition of the additive inverse:
[tex]\[
(6m^5 + 3 - m^3 - 4m) - (-m^5 + 2m^3 - 4m + 6)
\][/tex]
becomes
[tex]\[
(6m^5 + 3 - m^3 - 4m) + (m^5 - 2m^3 + 4m - 6)
\][/tex]
2. Rewrite terms that are subtracted as addition of the opposite:
[tex]\[
6m^5 + 3 + (-m^3) + (-4m) + m^5 + (-2m^3) + 4m + (-6)
\][/tex]
3. Group like terms:
[tex]\[
(6m^5 + m^5) + (3 + (-6)) + (-m^3 + -2m^3) + (-4m + 4m)
\][/tex]
4. Combine like terms:
- For [tex]\(m^5\)[/tex] terms: [tex]\(6m^5 + m^5 = 7m^5\)[/tex]
- For constants: [tex]\(3 + (-6) = -3\)[/tex]
- For [tex]\(m^3\)[/tex] terms: [tex]\(-m^3 + -2m^3 = -3m^3\)[/tex]
- For [tex]\(m\)[/tex] terms: [tex]\(-4m + 4m = 0\)[/tex]
5. Write the resulting polynomial in standard form:
[tex]\[
7m^5 - 3m^3 + 0m - 3
\][/tex]
So, the final result is:
[tex]\[
7m^5 - 3m^3 - 3
\][/tex]
Thus, the simplified polynomial is:
[tex]\[ \boxed{7m^5 - 3m^3 - 3} \][/tex]
