Sure, let's solve and simplify the given expression step-by-step.
Given the expression:
[tex]\[
\frac{2x - 1 - 4ax + 2a}{2x^2 - x}
\][/tex]
First, we'll simplify the numerator and the denominator separately where possible.
### Step 1: Simplify the numerator
The numerator is:
[tex]\[
2x - 1 - 4ax + 2a
\][/tex]
Group the terms involving [tex]\(x\)[/tex] together in the numerator:
[tex]\[
(2x - 4ax) + (-1 + 2a)
\][/tex]
Factor [tex]\(x\)[/tex] out from the first group:
[tex]\[
2x(1 - 2a) + (-1 + 2a)
\][/tex]
So, the numerator simplifies to:
[tex]\[
2x(1 - 2a) + (-1 + 2a)
\][/tex]
### Step 2: Simplify the denominator
The denominator is:
[tex]\[
2x^2 - x
\][/tex]
Factor out [tex]\(x\)[/tex] from the denominator:
[tex]\[
x(2x - 1)
\][/tex]
### Step 3: Combine and simplify the expression
Putting the simplified numerator and denominator together:
[tex]\[
\frac{2x(1 - 2a) + (-1 + 2a)}{x(2x - 1)}
\][/tex]
To simplify this expression, we need to separate the terms in the numerator and then try to cancel common factors. Split the fraction:
[tex]\[
= \frac{2x(1 - 2a)}{x(2x - 1)} + \frac{-1 + 2a}{x(2x - 1)}
\][/tex]
We can cancel [tex]\(x\)[/tex] in the first term of the numerator and the denominator:
[tex]\[
= \frac{2(1 - 2a)}{2x - 1} + \frac{-1 + 2a}{x(2x - 1)}
\][/tex]
Now observe that both parts individually do not share any further common factors, thus the final simplified version of the expression is:
[tex]\[
\frac{1 - 2a}{x}
\][/tex]
Hence, the expression simplifies to:
[tex]\[
\frac{1 - 2a}{x}
\][/tex]
