To find the result of the polynomial subtraction [tex]\(\left(6 m^5+3-m^3-4 m\right)-\left(-m^5+2 m^3-4 m+6\right)\)[/tex], let's follow the step-by-step process:
1. Write the subtraction of a polynomial expression as addition of the additive inverse:
[tex]\[
\left(6 m^5 + 3 - m^3 - 4 m\right) + \left(m^5 - 2 m^3 + 4 m - 6\right)
\][/tex]
2. Rewrite terms that are subtracted as addition of the opposite:
[tex]\[
6 m^5 + 3 + (-m^3) + (-4 m) + m^5 + (-2 m^3) + 4 m + (-6)
\][/tex]
3. Group like terms:
[tex]\[
\left[6 m^5 + m^5\right] + [3 + (-6)] + \left[(-m^3) + (-2 m^3)\right] + [(-4 m) + 4 m]
\][/tex]
4. Combine like terms:
- Combine the [tex]\(m^5\)[/tex] terms:
[tex]\[
6 m^5 + m^5 = 7 m^5
\][/tex]
- Combine the constant terms:
[tex]\[
3 + (-6) = 3 - 6 = -3
\][/tex]
- Combine the [tex]\(m^3\)[/tex] terms:
[tex]\[
-m^3 + (-2 m^3) = -m^3 - 2 m^3 = -3 m^3
\][/tex]
- Combine the [tex]\(m\)[/tex] terms (note that they cancel out):
[tex]\[
-4 m + 4 m = 0
\][/tex]
5. Write the resulting polynomial in standard form:
[tex]\[
7 m^5 - 3 m^3 - 3
\][/tex]
Therefore, the result of the given polynomial subtraction is:
[tex]\[
\boxed{7 m^5 - 3 m^3 - 3}
\][/tex]
