To find the product of the given expression:
[tex]\[
6 \left(x^2 - 1\right) \cdot \frac{6x - 1}{6(x + 1)}
\][/tex]
we'll simplify it step by step.
1. Factorization of [tex]\( x^2 - 1 \)[/tex]:
Notice that [tex]\( x^2 - 1 \)[/tex] can be factored using the difference of squares:
[tex]\[
x^2 - 1 = (x + 1)(x - 1)
\][/tex]
2. Formulate the expression:
Substituting the factorization into the given expression gives:
[tex]\[
6 \left[(x + 1)(x - 1)\right] \cdot \frac{6x - 1}{6(x + 1)}
\][/tex]
3. Cancel out common factors:
Now, we can cancel out the common factor [tex]\( x + 1 \)[/tex]:
[tex]\[
6 \left[(x + 1)(x - 1)\right] \cdot \frac{6x - 1}{6(x + 1)} = 6(x - 1) \cdot (6x - 1)
\][/tex]
4. Express the simplified product:
We now have the simplified product:
[tex]\[
6 \left(x - 1\right) \left(6x - 1\right)
\][/tex]
To further simplify:
5. Distribute the 6:
Let's expand the expression inside the parentheses:
[tex]\[
6 \left[(x - 1)(6x - 1)\right]
\][/tex]
6. Expand the polynomial:
Multiply the binomials:
[tex]\[
(x - 1)(6x - 1) = x \cdot 6x + x \cdot (-1) - 1 \cdot 6x - 1 \cdot -1
\][/tex]
Which simplifies to:
[tex]\[
(x - 1)(6x - 1) = 6x^2 - x - 6x + 1 = 6x^2 - 7x + 1
\][/tex]
7. Multiply by 6:
Hence, multiplying by 6 gives:
[tex]\[
6 \left(6x^2 - 7x + 1\right)
\][/tex]
Therefore, the complete and simplified product for the given expression is:
[tex]\[
6x^2 - 7x + 1
\][/tex]
