To simplify the given expression [tex]\((4 - 2x) \left(\frac{-6x^2 + 10x + 21}{4 - 2x}\right)\)[/tex], we follow these steps:
1. Identify the expression:
[tex]\[
(4 - 2x) \left( \frac{-6x^2 + 10x + 21}{4 - 2x} \right)
\][/tex]
2. Cancel the common terms in the numerator and the denominator:
Notice that the [tex]\((4 - 2x)\)[/tex] term in the numerator and the denominator of the fraction cancels each other out:
[tex]\[
(4 - 2x) \left( \frac{-6x^2 + 10x + 21}{4 - 2x} \right) = -6x^2 + 10x + 21
\][/tex]
This leaves us with the simplified expression:
[tex]\[
-6x^2 + 10x + 21
\][/tex]
3. Compare the simplified expression with the given options:
Now, we compare the obtained expression [tex]\(-6x^2 + 10x + 21\)[/tex] with the given options:
- [tex]\(-x^2 - 2x + 12\)[/tex]
- [tex]\(-x^2 + 2x - 8\)[/tex]
- [tex]\(-6x^2 + 10x + 21\)[/tex]
- [tex]\(-x^3 + 5x^2 - 6x + 3\)[/tex]
We observe that the simplified expression exactly matches the third option:
[tex]\[
-6x^2 + 10x + 21
\][/tex]
Thus, the result of the expression [tex]\((4 - 2x) \left(\frac{-6x^2 + 10x + 21}{4 - 2x}\right)\)[/tex] is [tex]\(-6x^2 + 10x + 21\)[/tex]. This matches the third given option.
So, the completely simplified form of the expression is:
[tex]\[
\boxed{-6x^2 + 10x + 21}
\][/tex]
