To solve the problem of distributing each term of the first polynomial [tex]\(\left(2x - x^3\right)\)[/tex] onto the second polynomial [tex]\(\left(-3x^4 - 7x^2\right)\)[/tex], let's do the distribution step-by-step.
1. Distribute [tex]\(2x\)[/tex] to each term in the second polynomial:
[tex]\[
2x \left(-3x^4 - 7x^2\right)
\][/tex]
2. Distribute [tex]\(-x^3\)[/tex] to each term in the second polynomial:
[tex]\[
-x^3 \left(-3x^4 - 7x^2\right)
\][/tex]
3. Combining the above steps, the distributed expression is:
[tex]\[
2x \left(-3x^4 - 7x^2\right) + (-x^3) \left(-3x^4 - 7x^2\right)
\][/tex]
Now, evaluate the answer choices:
- The first option:
[tex]\[
2 x \left(-3 x^4 - 7 x^2\right) - x^3 \left(-3 x^4 - 7 x^2\right)
\][/tex]
This correctly matches the combining step we did above. It distributes both [tex]\(2x\)[/tex] and [tex]\(-x^3\)[/tex] correctly to each term in [tex]\(-3x^4 - 7x^2\)[/tex].
- The second option:
[tex]\[
2 x \left(-3 x^4\right) - x^3 \left(-7 x^2\right)
\][/tex]
This does not correctly distribute each term in [tex]\(2x - x^3\)[/tex] to all terms in [tex]\(-3x^4 - 7x^2\)[/tex].
- The third option:
[tex]\[
2 x \left(-x^3 - 3 x^4 - 7 x^2\right)
\][/tex]
This introduces an incorrect term [tex]\(-x^3\)[/tex] within the parentheses, which is nowhere derived from the original expressions.
- The fourth option:
[tex]\[
2 x \left(-3 x^4 - 7 x^2\right) + x^3 \left(-3 x^4 - 7 x^2\right)
\][/tex]
This incorrectly changes the sign of the second term that involves [tex]\(-x^3\)[/tex].
Thus, the correct expression is:
[tex]\[
\boxed{2 x\left(-3 x^4-7 x^2\right)-x^3\left(-3 x^4-7 x^2\right)}
\][/tex]
