To find the factors of the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] by grouping, we can follow these steps:
1. Group the terms:
[tex]\[
12x^3 - 2x^2 + 18x - 3 = (12x^3 + 18x) - (2x^2 + 3)
\][/tex]
2. Factor out the greatest common factor (GCF) from each group:
- For the first group, [tex]\(12x^3 + 18x\)[/tex]:
[tex]\[
12x^3 + 18x = 6x(2x^2 + 3)
\][/tex]
- For the second group, [tex]\(-(2x^2 + 3)\)[/tex]:
[tex]\[
-(2x^2 + 3) = -1(2x^2 + 3)
\][/tex]
3. Rewrite the expression with these factorizations:
[tex]\[
12x^3 - 2x^2 + 18x - 3 = 6x(2x^2 + 3) - 1(2x^2 + 3)
\][/tex]
4. Factor out the common binomial factor [tex]\((2x^2 + 3)\)[/tex]:
[tex]\[
12x^3 - 2x^2 + 18x - 3 = (6x - 1)(2x^2 + 3)
\][/tex]
Therefore, the polynomial [tex]\(12x^3 - 2x^2 + 18x - 3\)[/tex] is factored as:
[tex]\[
6x(2x^2 + 3) + 1(2x^2 + 3)
\][/tex]
So, the correct choice is:
[tex]\[
\boxed{6x\left(2x^2+3\right)+1\left(2x^2+3\right)}
\][/tex]
