To find the product of the given expression:
[tex]\[
6 \left(x^2 - 1\right) \cdot \frac{6x - 1}{6(x + 1)}
\][/tex]
Here are the step-by-step details:
1. Expand and Simplify the Expression:
The first part of the expression is:
[tex]\[
6 \left(x^2 - 1\right)
\][/tex]
Recall that [tex]\( x^2 - 1 \)[/tex] can be factored as:
[tex]\[
x^2 - 1 = (x - 1)(x + 1)
\][/tex]
So, we have:
[tex]\[
6 \left(x^2 - 1\right) = 6 (x - 1)(x + 1)
\][/tex]
2. Multiply the two expressions:
Now, multiply this with the second part of the expression:
[tex]\[
6 (x - 1)(x + 1) \cdot \frac{6x - 1}{6(x + 1)}
\][/tex]
3. Simplify by Cancelling Common Factors:
Notice that [tex]\( 6(x + 1) \)[/tex] appears in both the numerator and the denominator, so they can be canceled:
[tex]\[
6 (x - 1)(x + 1) \cdot \frac{6x - 1}{6(x + 1)} = (6 (x - 1) (x + 1)) \cdot \frac{6x - 1}{6(x + 1)} = (6 (x - 1) \cdot (x + 1) \cdot (6x - 1)) / (6 (x + 1))
\][/tex]
Cancel [tex]\( x + 1 \)[/tex]:
[tex]\[
= (6(x - 1)(6x - 1)) / 6
\][/tex]
Simplify further by canceling the 6 in the numerator and denominator:
[tex]\[
(x - 1)(6x - 1)
\][/tex]
4. Distribute the Terms:
Now distribute [tex]\( x - 1 \)[/tex] through [tex]\( 6x - 1 \)[/tex]:
[tex]\[
(x - 1)(6x - 1) = x(6x - 1) - 1(6x - 1) = 6x^2 - x - 6x + 1 = 6x^2 - 7x + 1
\][/tex]
Therefore, the simplified product is:
[tex]\[
6x^2 - 7x + 1
\][/tex]
