Alright, let's go through the step-by-step process of factoring the polynomial [tex]\(3x^3 - 15x^2 + 8x - 40\)[/tex].
1. Group the terms in pairs:
[tex]\[
3x^3 - 15x^2 + 8x - 40 \Rightarrow (3x^3 - 15x^2) + (8x - 40)
\][/tex]
2. Factor out the greatest common factor from each pair:
- For the first group, [tex]\(3x^3 - 15x^2\)[/tex]:
[tex]\[
3x^2(x - 5)
\][/tex]
- For the second group, [tex]\(8x - 40\)[/tex]:
[tex]\[
8(x - 5)
\][/tex]
So we have:
[tex]\[
3x^2(x - 5) + 8(x - 5)
\][/tex]
3. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
Notice that [tex]\((x - 5)\)[/tex] is a common factor in both terms. Hence, we can factor it out:
[tex]\[
(x - 5)(3x^2 + 8)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(3x^3 - 15x^2 + 8x - 40\)[/tex] is:
[tex]\[
(x - 5)(3x^2 + 8)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{(3x^2 + 8)(x - 5)}
\][/tex]
