Let's solve the given problem step-by-step.
Given:
[tex]\[
\left(6 m^5+3-m^3-4 m\right)-\left(-m^5+2 m^3-4 m+6\right)
\][/tex]
1. Write the subtraction of the polynomial expression as the 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 the 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]\[
\Big[6 m^5 + m^5\Big] + \Big[3 + (-6)\Big] + \Big[(-m^3) + (-2 m^3)\Big] + \Big[(-4 m) + 4 m\Big]
\][/tex]
4. Combine like terms:
[tex]\[
(6 m^5 + m^5) + (3 - 6) + (-m^3 - 2 m^3) + (-4 m + 4 m)
\][/tex]
Calculate each group:
[tex]\[
(6 m^5 + m^5) = 7 m^5
\][/tex]
[tex]\[
(3 - 6) = -3
\][/tex]
[tex]\[
(-m^3 - 2 m^3) = -3 m^3
\][/tex]
[tex]\[
(-4 m + 4 m) = 0
\][/tex]
5. Write the resulting polynomial in standard form:
[tex]\[
7 m^5 - 3 m^3 + 0 m - 3
\][/tex]
Thus:
[tex]\[
\boxed{7 m^5 - 3 m^3 + 0 m - 3}
\][/tex]
Which simplifies to:
[tex]\[
7 m^5 - 3 m^3 - 3
\][/tex]
