Let's simplify the given mathematical expression step by step:
Given expression:
[tex]\[
6\left(x^2-1\right) \cdot \frac{6x-1}{6(x+1)}
\][/tex]
### Step 1: Factorize and Simplify
First, let's analyze each part of the expression separately.
#### Expression \(6(x^2 - 1)\):
The term \(x^2 - 1\) is a difference of squares, which can be factored as:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
So:
[tex]\[
6(x^2 - 1) = 6(x - 1)(x + 1)
\][/tex]
#### Expression \(\frac{6x-1}{6(x+1)}\):
This part is already in a simplified fractional form.
### Step 2: Combine the Expressions
Now we combine the two expressions:
[tex]\[
6(x - 1)(x + 1) \cdot \frac{6x - 1}{6(x + 1)}
\][/tex]
We can cancel out the common factor \((x + 1)\) in the numerator and the denominator:
[tex]\[
6(x - 1)(x + 1) \cdot \frac{6x - 1}{6(x + 1)} = 6(x - 1) \cdot \frac{6x - 1}{1} = 6(x - 1)(6x - 1)
\][/tex]
### Step 3: Check the Possible Options
Now we need to see which of the given options matches the simplified product:
1. \(6(x - 1)^2\)
2. \(6(x^2 - 1)\)
3. \((x + 1)(6x - 1)\)
4. \((x - 1)(6x - 1)\)
From our simplification, we see that the expression \(6(x - 1)(6x - 1)\) resembles option 4:
[tex]\[
(x - 1)(6x - 1)
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
