Bot GPT 4-o 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.
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