Sure, let's solve the given polynomial subtraction 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]
### Step 1: Write subtraction of a polynomial expression as addition of the additive inverse.
Rewriting subtraction as addition, we get:
[tex]\[
\left(6 m^5+3-m^3-4 m\right) + \left(m^5 - 2 m^3 + 4 m - 6\right)
\][/tex]
### Step 2: Rewrite terms that are subtracted as addition of the opposite.
So, we have:
[tex]\[
6 m^5 + 3 + (-m^3) + (-4 m) + m^5 + (-2 m^3) + 4 m + (-6)
\][/tex]
### Step 3: Group like terms.
Now, group the coefficients of similar powers of [tex]\(m\)[/tex]:
[tex]\[
[6 m^5 + m^5] + [3 + (-6)] + [(-m^3) + (-2 m^3)] + [(-4 m) + 4 m]
\][/tex]
### Step 4: Combine like terms.
Combine the coefficients for each power of [tex]\(m\)[/tex]:
1. Combine the [tex]\(m^5\)[/tex] terms:
[tex]\[
6 m^5 + m^5 = 7 m^5
\][/tex]
2. Combine the constant terms:
[tex]\[
3 + (-6) = -3
\][/tex]
3. Combine the [tex]\(m^3\)[/tex] terms:
[tex]\[
(-m^3) + (-2 m^3) = -3 m^3
\][/tex]
4. Combine the [tex]\(m\)[/tex] terms:
[tex]\[
(-4 m) + 4 m = 0 m
\][/tex]
### Step 5: Write the resulting polynomial in standard form.
After combining like terms, the resulting polynomial is:
[tex]\[
7 m^5 - 3 m^3 + 0 m - 3
\][/tex]
Therefore, the final answer in standard form is:
[tex]\[
7 m^5 - 3 m^3 - 3
\][/tex]
In conclusion, the coefficients you asked for are:
- Coefficient of [tex]\(m^5\)[/tex]: [tex]\(7\)[/tex]
- Coefficient of [tex]\(m^3\)[/tex]: [tex]\(-3\)[/tex]
- Coefficient of [tex]\(m\)[/tex]: [tex]\(0\)[/tex]
- Constant term: [tex]\(-3\)[/tex]
