Sure, let's simplify the given expression step by step. We have:
[tex]\[
\frac{\left(2 x^5 y^7\right)\left(-5 x^4 z^{-5}\right)}{y^{12} z^6}
\][/tex]
Step 1: Simplify the numerator.
First, multiply the terms in the numerator:
[tex]\[
\left(2 x^5 y^7\right)\left(-5 x^4 z^{-5}\right)
\][/tex]
To multiply these, combine the coefficients and the variables with the same base:
[tex]\[
2 \cdot -5 = -10
\][/tex]
For the exponents, we add the exponents of the same variables:
[tex]\[
x^{5+4} = x^9
\][/tex]
[tex]\[
y^{7}
\][/tex]
[tex]\[
z^{-5}
\][/tex]
So, the simplified numerator is:
[tex]\[
-10 x^9 y^7 z^{-5}
\][/tex]
Step 2: Combine with the denominator.
Next, we divide the simplified numerator by the denominator:
[tex]\[
\frac{-10 x^9 y^7 z^{-5}}{y^{12} z^6}
\][/tex]
For the variables in the numerator and the denominator, we subtract the exponents of the same bases:
[tex]\[
y^{7-12} = y^{-5}
\][/tex]
[tex]\[
z^{-5-6} = z^{-11}
\][/tex]
So, now we have:
[tex]\[
-10 x^9 y^{-5} z^{-11}
\][/tex]
Step 3: Convert negative exponents to positive by moving them to the denominator.
[tex]\[
y^{-5} = \frac{1}{y^5}
\][/tex]
[tex]\[
z^{-11} = \frac{1}{z^{11}}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{-10 x^9}{y^5 z^{11}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-\frac{10 x^9}{y^5 z^{11}}}
\][/tex]
From the given options, this matches:
D. [tex]\(\boxed{-\frac{10 x^9}{y^5 z^{11}}}\)[/tex]
