To solve the given expression:
[tex]\[ 6\left(x^2 - 1\right) \cdot \frac{6x - 1}{6(x + 1)} \][/tex]
let's break it down step-by-step.
### Step-by-Step Solution:
1. Factorize [tex]\( x^2 - 1 \)[/tex]:
Notice that [tex]\( x^2 - 1 \)[/tex] can be factorized using the difference of squares:
[tex]\[ x^2 - 1 = (x - 1)(x + 1) \][/tex]
2. Substitute the factorized form:
Now rewrite the expression by replacing [tex]\( x^2 - 1 \)[/tex]:
[tex]\[ 6\left( (x - 1)(x + 1) \right) \cdot \frac{6x - 1}{6(x + 1)} \][/tex]
3. Simplify the expression:
Next, observe that we have [tex]\( (x + 1) \)[/tex] in both the numerator and the denominator:
[tex]\[ 6(x - 1)(x + 1) \cdot \frac{6x - 1}{6(x + 1)} \][/tex]
The [tex]\( (x + 1) \)[/tex] terms cancel out:
[tex]\[ 6(x - 1) \cdot \frac{6x - 1}{6} \][/tex]
4. Further simplification:
Now notice that there's a factor of 6 in the numerator and denominator which cancels out:
[tex]\[ (x - 1) \cdot (6x - 1) \][/tex]
So, the product of the given expression is:
[tex]\[ (x - 1)(6x - 1) \][/tex]
Hence, the correct answer is:
[tex]\[(x-1)(6x-1)\][/tex]
