Sure, let's break down the given expression step-by-step so we understand it thoroughly. The expression we need to work with is:
[tex]\[ \frac{2x - 1}{(x + 1)(3x - 5)^2} \][/tex]
### Step 1: Identify the Components
First, let’s identify the components of the expression. We have a numerator and a denominator.
- Numerator: [tex]\( 2x - 1 \)[/tex]
- Denominator: [tex]\((x + 1)(3x - 5)^2\)[/tex]
### Step 2: Examine the Numerator
The numerator is a linear polynomial:
[tex]\[ 2x - 1 \][/tex]
### Step 3: Examine the Denominator
The denominator is a product of two expressions:
1. [tex]\( x + 1 \)[/tex]
2. [tex]\( (3x - 5)^2 \)[/tex]
Putting these together, the denominator becomes:
[tex]\[ (x + 1) \cdot (3x - 5)^2 \][/tex]
### Step 4: Combine Numerator and Denominator
After identifying the components, we combine the numerator and the denominator to form the given fraction:
[tex]\[ \frac{2x - 1}{(x + 1)(3x - 5)^2} \][/tex]
### Summary
By following the breakdown steps and combining the elements, we confirm that our final expression is:
[tex]\[ \frac{2x - 1}{(x + 1)(3x - 5)^2} \][/tex]
This given expression represents our complete and final answer as simplified. No further simplification of this rational function is possible without additional context or constraints.
