Perform the operation(s) and simplify completely.

Question

(4−2x)(−6x2+10x+214−2x)


(1 point)
Responses

−x2−2x+12
negative x squared minus 2 x plus 12

−x3+5x2−6x+3
negative x cubed plus 5 x squared minus 6 x plus 3

−6x2+10x+21
negative 6 x squared plus 10 x plus 21

−x2+2x−8

Answer 1

Let's simplify the expression \((4−2x)(−6x^2+10x+21−2x)\).

First, we simplify the second factor:

\[ -6x^2 + 10x + 21 - 2x = -6x^2 + (10x - 2x) + 21 = -6x^2 + 8x + 21 \]

Now, we rewrite the expression:

\[ (4 - 2x)(-6x^2 + 8x + 21) \]

Next, we can perform the multiplication using the distributive property (also known as the FOIL method for binomials):

\[ = 4(-6x^2) + 4(8x) + 4(21) - 2x(-6x^2) - 2x(8x) - 2x(21) \]

Calculating each term:

1. \(4(-6x^2) = -24x^2\)
2. \(4(8x) = 32x\)
3. \(4(21) = 84\)
4. \(-2x(-6x^2) = 12x^3\)
5. \(-2x(8x) = -16x^2\)
6. \(-2x(21) = -42x\)

Now, combine all these terms together:

\[ 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84 \]

Simplifying the coefficients:

1. \(12x^3\)
2. \(-24x^2 - 16x^2 = -40x^2\)
3. \(32x - 42x = -10x\)
4. Constant term remains \(84\)

Putting it all together:

\[ 12x^3 - 40x^2 - 10x + 84 \]

Therefore, the simplified expression for \((4−2x)(−6x^2+10x+21−2x)\) is:

\[ 12x^3 - 40x^2 - 10x + 84 \]

From the provided options, none of them match. Based on our calculations, the result is:

\[ 12x^3 - 40x^2 - 10x + 84 \]

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