To determine which expression shows one way to factorize the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping, we will evaluate each option to see which correctly simplifies back to the given polynomial.
### Option 1: [tex]\(2x^2(6x - 1) + 3(6x - 1)\)[/tex]
Let's expand this:
[tex]\[
2x^2(6x - 1) + 3(6x - 1) = 2x^2 \cdot 6x + 2x^2 \cdot (-1) + 3 \cdot 6x + 3 \cdot (-1)
\][/tex]
[tex]\[
= 12x^3 - 2x^2 + 18x - 3
\][/tex]
This expansion matches the given polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex]. So, Option 1 correctly factorizes the polynomial through grouping.
### Option 2: [tex]\(2x^2(6x - 1) - 3(6x - 1)\)[/tex]
Let's expand this:
[tex]\[
2x^2(6x - 1) - 3(6x - 1) = 2x^2 \cdot 6x + 2x^2 \cdot (-1) - 3 \cdot 6x - 3 \cdot (-1)
\][/tex]
[tex]\[
= 12x^3 - 2x^2 - 18x + 3
\][/tex]
This expansion gives [tex]\(12x^3 - 2x^2 - 18x + 3\)[/tex], which does not match the given polynomial. Therefore, Option 2 is incorrect.
### Option 3: [tex]\(6x(2x^2 - 3) - 1(2x^2 - 3)\)[/tex]
Let's expand this:
[tex]\[
6x(2x^2 - 3) - 1(2x^2 - 3) = 6x \cdot 2x^2 + 6x \cdot (-3) - 1 \cdot 2x^2 - 1 \cdot (-3)
\][/tex]
[tex]\[
= 12x^3 - 18x - 2x^2 + 3
\][/tex]
This expansion gives [tex]\(12x^3 - 18x - 2x^2 + 3\)[/tex], which does not match the given polynomial. Therefore, Option 3 is incorrect.
### Option 4: [tex]\(6x(2x^2 + 3) + 1(2x^2 + 3)\)[/tex]
Let's expand this:
[tex]\[
6x(2x^2 + 3) + 1(2x^2 + 3) = 6x \cdot 2x^2 + 6x \cdot 3 + 1 \cdot 2x^2 + 1 \cdot 3
\][/tex]
[tex]\[
= 12x^3 + 18x + 2x^2 + 3
\][/tex]
This expansion gives [tex]\(12x^3 + 18x + 2x^2 + 3\)[/tex], which does not match the given polynomial. Therefore, Option 4 is incorrect.
Thus, the correct way to determine one way to factorize [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping is:
[tex]\[
\boxed{2x^2(6x - 1) + 3(6x - 1)}
\][/tex]
