To simplify the expression \((4 - 2x)(-6x^2 + 10x + \frac{21}{4 - 2x})\), let's first observe how the terms interact with each other.
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Simplifying the second term:
- The term \(\frac{21}{4 - 2x}\) can be seen as a division and will not interact directly as a polynomial. We can look at the expression more closely.
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Multiplying the two terms:
- We first rewrite the expression without the fraction: \[ (4 - 2x)(-6x^2 + 10x + \frac{21}{4 - 2x}) \]
We separate it into two parts: \[ (4 - 2x)(-6x^2 + 10x) + (4 - 2x)\frac{21}{4 - 2x} \]
The second part simplifies to \( 21 \): \[ 4 - 2x \text{ cancels with the denominator in that part.} \]
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Distributing the first part: Now, we will focus on simplifying: \[ (4 - 2x)(-6x^2 + 10x) \] By distributing: \[ = 4(-6x^2) + 4(10x) - 2x(-6x^2) - 2x(10x) \] \[ = -24x^2 + 40x + 12x^3 - 20x^2 \] Combining like terms: \[ 12x^3 - 24x^2 - 20x^2 + 40x = 12x^3 - 44x^2 + 40x \]
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Combining everything: Adding \(21\) to this gives us: \[ 12x^3 - 44x^2 + 40x + 21 \]
Thus, the final simplified expression is: \[ 12x^3 - 44x^2 + 40x + 21 \]
None of the provided responses match this solution directly. If we assume some simplification or if the problem was slightly different to allow a match, we need to check for any mistakes. However, from following the proper steps: The closest option could be chosen as it appears to match patterns, but strictly the calculation performed doesn’t yield one of the provided options.
So since the provided responses do not match this result, the answer might either have a mistake in the question or the options need to be verified.
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