Sure, let's solve the division problem step-by-step.
We need to simplify the expression:
[tex]\[ \frac{20 - 15m + 5m^3 - m^5}{-5m} \][/tex]
Let's divide each term in the numerator by [tex]\(-5m\)[/tex].
1. Divide [tex]\(20\)[/tex] by [tex]\(-5m\)[/tex]:
[tex]\[ \frac{20}{-5m} = -\frac{20}{5m} = -\frac{4}{m} \][/tex]
2. Divide [tex]\(-15m\)[/tex] by [tex]\(-5m\)[/tex]:
[tex]\[ \frac{-15m}{-5m} = \frac{15m}{5m} = 3 \][/tex]
3. Divide [tex]\(5m^3\)[/tex] by [tex]\(-5m\)[/tex]:
[tex]\[ \frac{5m^3}{-5m} = -\frac{5m^3}{5m} = -m^2 \][/tex]
4. Finally, divide [tex]\(-m^5\)[/tex] by [tex]\(-5m\)[/tex]:
[tex]\[ \frac{-m^5}{-5m} = \frac{m^5}{5m} = \frac{m^4}{5} \][/tex]
Now, combine all these results together:
[tex]\[ -\frac{4}{m} + 3 - m^2 + \frac{m^4}{5} \][/tex]
Rewriting in a simpler form, we get:
[tex]\[ \frac{m^4}{5} - m^2 + 3 - \frac{4}{m} \][/tex]
Thus, the final simplified result of the division is:
[tex]\[ \boxed{\frac{m^4}{5} - m^2 + 3 - \frac{4}{m}} \][/tex]
