Question
Perform the operation(s) and simplify completely.
(4−2x)(−6x2+10x+214−2x)
(1 point)
Responses
−6x2+10x+21
negative 6 x squared plus 10 x plus 21
−x2+2x−8
negative x squared plus 2 x minus 8
−x3+5x2−6x+3
negative x cubed plus 5 x squared minus 6 x plus 3
−x2−2x+12
(4−2x)(−6x2+10x+214−2x)
(1 point)
Responses
−6x2+10x+21
negative 6 x squared plus 10 x plus 21
−x2+2x−8
negative x squared plus 2 x minus 8
−x3+5x2−6x+3
negative x cubed plus 5 x squared minus 6 x plus 3
−x2−2x+12
Answers
GPT-4o mini
To simplify the expression \((4−2x)(−6x^2+10x+21−2x)\), we first simplify what is inside the parentheses.
Combining like terms:
\[
−6x^2 + 10x + 21 - 2x = −6x^2 + (10x - 2x) + 21 = −6x^2 + 8x + 21
\]
Now we have:
\[
(4−2x)(−6x^2 + 8x + 21)
\]
Next, we will use the distributive property (also known as the FOIL method for binomials) to expand this expression:
\[
= 4 \cdot (−6x^2 + 8x + 21) + (−2x) \cdot (−6x^2 + 8x + 21)
\]
Calculating the first part:
\[
4 \cdot (−6x^2) = -24x^2
\]
\[
4 \cdot 8x = 32x
\]
\[
4 \cdot 21 = 84
\]
So, the first part results in:
\[
-24x^2 + 32x + 84
\]
Now for the second part:
\[
(−2x) \cdot (−6x^2) = 12x^3
\]
\[
(−2x) \cdot 8x = -16x^2
\]
\[
(−2x) \cdot 21 = -42x
\]
So, the second part results in:
\[
12x^3 - 16x^2 - 42x
\]
Now we combine all the terms:
\[
12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84
\]
Combine like terms:
\[
= 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84
= 12x^3 - 40x^2 - 10x + 84
\]
This gives us the final simplified expression:
\[
\boxed{12x^3 - 40x^2 - 10x + 84}
\]
The values provided do not match, but the simplified final expression is \(12x^3 - 40x^2 - 10x + 84\). If this does not match perfectly with the responses, it may suggest an issue with the initial expression interpretation or simplification.
Combining like terms:
\[
−6x^2 + 10x + 21 - 2x = −6x^2 + (10x - 2x) + 21 = −6x^2 + 8x + 21
\]
Now we have:
\[
(4−2x)(−6x^2 + 8x + 21)
\]
Next, we will use the distributive property (also known as the FOIL method for binomials) to expand this expression:
\[
= 4 \cdot (−6x^2 + 8x + 21) + (−2x) \cdot (−6x^2 + 8x + 21)
\]
Calculating the first part:
\[
4 \cdot (−6x^2) = -24x^2
\]
\[
4 \cdot 8x = 32x
\]
\[
4 \cdot 21 = 84
\]
So, the first part results in:
\[
-24x^2 + 32x + 84
\]
Now for the second part:
\[
(−2x) \cdot (−6x^2) = 12x^3
\]
\[
(−2x) \cdot 8x = -16x^2
\]
\[
(−2x) \cdot 21 = -42x
\]
So, the second part results in:
\[
12x^3 - 16x^2 - 42x
\]
Now we combine all the terms:
\[
12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84
\]
Combine like terms:
\[
= 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84
= 12x^3 - 40x^2 - 10x + 84
\]
This gives us the final simplified expression:
\[
\boxed{12x^3 - 40x^2 - 10x + 84}
\]
The values provided do not match, but the simplified final expression is \(12x^3 - 40x^2 - 10x + 84\). If this does not match perfectly with the responses, it may suggest an issue with the initial expression interpretation or simplification.