To factor the polynomial [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] by grouping, we can proceed with the following steps:
1. Group the polynomial into two pairs:
[tex]\[
(15x^3 - 5x^2) + (6x - 2)
\][/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- For the first group, [tex]\(15x^3 - 5x^2\)[/tex], the GCF is [tex]\(5x^2\)[/tex]:
[tex]\[
15x^3 - 5x^2 = 5x^2(3x - 1)
\][/tex]
- For the second group, [tex]\(6x - 2\)[/tex], the GCF is 2:
[tex]\[
6x - 2 = 2(3x - 1)
\][/tex]
3. Rewrite the polynomial using these common factors:
[tex]\[
15x^3 - 5x^2 + 6x - 2 = 5x^2(3x - 1) + 2(3x - 1)
\][/tex]
4. Notice that [tex]\((3x - 1)\)[/tex] is a common factor in both terms:
[tex]\[
= (3x - 1)(5x^2 + 2)
\][/tex]
Therefore, the factored form of the polynomial [tex]\(15x^3 - 5x^2 + 6x - 2\)[/tex] is:
[tex]\[
(3x - 1)(5x^2 + 2)
\][/tex]
Thus, the correct choice is:
[tex]\[
\left(5x^2 + 2\right)(3x - 1)
\][/tex]
