Certainly! Let's simplify the given expression step-by-step.
The expression we are given is:
[tex]\[
\frac{1}{2 x^2 - 4 x} - \frac{2}{x}
\][/tex]
First, factor the denominator of the first term:
[tex]\[
2 x^2 - 4 x = 2 x (x - 2)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{1}{2 x (x - 2)} - \frac{2}{x}
\][/tex]
Next, we need to have a common denominator to combine these fractions. The common denominator for [tex]\( x \)[/tex] and [tex]\( 2 x (x - 2) \)[/tex] is [tex]\( 2 x (x - 2) \)[/tex].
Rewriting the fractions with this common denominator:
[tex]\[
\frac{1}{2 x (x - 2)} - \frac{2 \cdot (2 (x - 2))}{2 x (x - 2)}
\][/tex]
Simplify the second term:
[tex]\[
\frac{1}{2 x (x - 2)} - \frac{4 (x - 2)}{2 x (x - 2)}
\][/tex]
Combine the fractions:
[tex]\[
\frac{1 - 4 (x - 2)}{2 x (x - 2)}
\][/tex]
Distribute the 4 in the numerator:
[tex]\[
1 - 4 x + 8 = 9 - 4 x
\][/tex]
So, the expression is:
[tex]\[
\frac{9 - 4 x}{2 x (x - 2)}
\][/tex]
This matches choice [tex]\( D \)[/tex]:
[tex]\[
D. \frac{-4 x + 9}{2 x (x - 2)}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
