To distribute the first polynomial \( (2x - x^3) \) onto the second polynomial \( (-3x^4 - 7x^2) \), we need to take each term from the first polynomial and multiply it by every term from the second polynomial.
This can be done as follows:
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Distributing \( 2x \) over \( (-3x^4 - 7x^2) \): \[ 2x \cdot (-3x^4) + 2x \cdot (-7x^2) = -6x^5 - 14x^3 \]
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Distributing \( -x^3 \) over \( (-3x^4 - 7x^2) \): \[ -x^3 \cdot (-3x^4) + -x^3 \cdot (-7x^2) = 3x^7 + 7x^5 \]
Combining these gives: \[ -6x^5 - 14x^3 + 3x^7 + 7x^5 = 3x^7 + (-6x^5 + 7x^5) - 14x^3 = 3x^7 + x^5 - 14x^3 \]
Now, to determine which of the provided expressions matches this distribution:
The closest expression is: \[ 2x(-3x^4 - 7x^2) + x^3(-3x^4 - 7x^2) \]
This represents correctly distributing each term of the first polynomial onto the second.
So the correct choice is: \[ 2x(-3x^4 - 7x^2) + x^3(-3x^4 - 7x^2) \]