Alright, let's simplify the given expression step by step:
[tex]\[
\frac{-20 z^3 u + 28 z^7 u^6 - 12 z^6 u^5}{4 z^5 u^4}
\][/tex]
We will divide each term in the numerator by the denominator separately.
1. First Term:
[tex]\[
\frac{-20 z^3 u}{4 z^5 u^4}
\][/tex]
Simplify the coefficients:
[tex]\[
\frac{-20}{4} = -5
\][/tex]
Simplify the [tex]\(z\)[/tex] terms:
[tex]\[
\frac{z^3}{z^5} = \frac{1}{z^2} = z^{-2}
\][/tex]
Simplify the [tex]\(u\)[/tex] terms:
[tex]\[
\frac{u}{u^4} = \frac{1}{u^3} = u^{-3}
\][/tex]
So, the first term simplifies to:
[tex]\[
-5 z^{-2} u^{-3}
\][/tex]
2. Second Term:
[tex]\[
\frac{28 z^7 u^6}{4 z^5 u^4}
\][/tex]
Simplify the coefficients:
[tex]\[
\frac{28}{4} = 7
\][/tex]
Simplify the [tex]\(z\)[/tex] terms:
[tex]\[
\frac{z^7}{z^5} = z^2
\][/tex]
Simplify the [tex]\(u\)[/tex] terms:
[tex]\[
\frac{u^6}{u^4} = u^2
\][/tex]
So, the second term simplifies to:
[tex]\[
7 z^2 u^2
\][/tex]
3. Third Term:
[tex]\[
\frac{-12 z^6 u^5}{4 z^5 u^4}
\][/tex]
Simplify the coefficients:
[tex]\[
\frac{-12}{4} = -3
\][/tex]
Simplify the [tex]\(z\)[/tex] terms:
[tex]\[
\frac{z^6}{z^5} = z
\][/tex]
Simplify the [tex]\(u\)[/tex] terms:
[tex]\[
\frac{u^5}{u^4} = u
\][/tex]
So, the third term simplifies to:
[tex]\[
-3 z u
\][/tex]
Combining all the simplified terms, the entire expression simplifies to:
[tex]\[
-5 z^{-2} u^{-3} + 7 z^2 u^2 - 3 z u
\][/tex]
Rewriting with positive exponents where possible, we get:
[tex]\[
\boxed{- \frac{5}{z^2 u^3} + 7 z^2 u^2 - 3 z u}
\][/tex]
