Let's determine the factors of the given polynomial [tex]\( 4x^3 + x^2 - 8x - 2 \)[/tex] by grouping.
1. Given polynomial:
[tex]\[
4x^3 + x^2 - 8x - 2
\][/tex]
2. Group terms to facilitate factoring:
[tex]\[
(4x^3 + x^2) + (-8x - 2)
\][/tex]
3. Factor out the greatest common factor from each group:
- For the first group, [tex]\( 4x^3 + x^2 \)[/tex]:
[tex]\[
x^2(4x + 1)
\][/tex]
- For the second group, [tex]\( -8x - 2 \)[/tex]:
[tex]\[
-2(4x + 1)
\][/tex]
4. Now, we can rewrite the polynomial:
[tex]\[
x^2(4x + 1) - 2(4x + 1)
\][/tex]
5. Notice that [tex]\( 4x + 1 \)[/tex] is a common factor:
[tex]\[
(4x + 1)(x^2 - 2)
\][/tex]
So, the polynomial [tex]\( 4x^3 + x^2 - 8x - 2 \)[/tex] is factored by grouping as:
[tex]\[
x^2(4x + 1) - 2(4x + 1)
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x^2(4x + 1) - 2(4x + 1)}
\][/tex]
