To simplify the expression \((4-2x) \left(-\frac{6x^2 + 10x + 21}{4-2x}\right)\), we can first notice that \(4 - 2x\) is a common factor in the numerator and the denominator.
We can rewrite the expression as follows:
\[ (4-2x) \left(-\frac{6x^2 + 10x + 21}{4-2x}\right) = - (6x^2 + 10x + 21) \]
This simplifies to:
\[
- (6x^2 + 10x + 21) = -6x^2 - 10x - 21 \]
Thus, the final expression is:
\[ -6x^2 - 10x - 21 \]
Now we need to check if any of the given options match this result. None of the options provided exactly match, however, we can check if \( - (6x^2 + 10x + 21) \) can still lead to a equivalent option:
Rewriting \(-6x^2 - 10x - 21\):
Now from the options:
A. \(-x^3 + 5x^2 - 6x + 3\)
B. \(-x^2 + 2x - 8\)
C. \(-6x^2 + 10x + 21\) (This is simply \(-6\) times what we have, not a match.)
D. \(-x^2 - 2x + 12\)
None of the options provided directly match the simplification.
So we double-check:
- Start again with the original:
\((4-2x) \left(-\frac{6x^2 + 10x + 21}{4-2x}\right)\)
Cancelling \((4 - 2x)\):
Which gives us \(- (6x^2 + 10x + 21)\), keeping an eye for correct signs, yield the same result.
Thus, the closest to my result is \(C\) but not quite equal, so make sure of values inserted:
After testing back:
None of the options fit. The simplification yields \(-6x^2 - 10x - 21\).
Final result doesn't match any of those provided correctly.
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