To simplify the expression [tex]\(\left(3 x^2 - 3\right)\left(2 x^3 - 8\right)\)[/tex], we will use the distributive property (or the FOIL method for binomials). Let's expand and simplify the expression step-by-step.
Given expression:
[tex]\[
(3 x^2 - 3)(2 x^3 - 8)
\][/tex]
Step 1: Distribute each term in the first binomial by each term in the second binomial.
[tex]\[
(3 x^2) \cdot (2 x^3) + (3 x^2) \cdot (-8) + (-3) \cdot (2 x^3) + (-3) \cdot (-8)
\][/tex]
Step 2: Perform the multiplications.
[tex]\[
3 x^2 \cdot 2 x^3 = 6 x^5
\][/tex]
[tex]\[
3 x^2 \cdot -8 = -24 x^2
\][/tex]
[tex]\[
-3 \cdot 2 x^3 = -6 x^3
\][/tex]
[tex]\[
-3 \cdot -8 = 24
\][/tex]
Step 3: Combine all the terms.
[tex]\[
6 x^5 - 24 x^2 - 6 x^3 + 24
\][/tex]
Step 4: Reorder the terms to match the format in the provided options, typically by descending powers of [tex]\(x\)[/tex]:
[tex]\[
6 x^5 - 6 x^3 - 24 x^2 + 24
\][/tex]
Finally, identify the simplified expression with the given options:
- [tex]\(6 x^5 - 30 x^2 - 24\)[/tex]
- [tex]\(6 x^5 - 30 x^2 + 24\)[/tex]
- [tex]\(6 x^5 - 6 x^3 - 24 x^2 + 24\)[/tex]
- [tex]\(6 x^6 - 6 x^3 - 24 x^2 + 24\)[/tex]
- [tex]\(6 x^5 - 6 x^3 - 24 x^2 - 24\)[/tex]
The correct simplified expression is:
[tex]\[
6 x^5 - 6 x^3 - 24 x^2 + 24
\][/tex]
Thus, we select:
[tex]\[
\boxed{6 x^5 - 6 x^3 - 24 x^2 + 24}
\][/tex]
