To simplify the expression \((4 - 2x)(-6x^2 + 10x + 21 - 2x)\), let's first simplify what we have inside the parentheses.
Combining like terms inside the second parentheses: \[ -6x^2 + 10x + 21 - 2x = -6x^2 + (10x - 2x) + 21 = -6x^2 + 8x + 21 \]
Now we rewrite the expression as: \[ (4 - 2x)(-6x^2 + 8x + 21) \]
Next, we'll use the distributive property (FOIL method) to multiply the two binomials.
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Multiply \(4\) by each term in the polynomial \(-6x^2 + 8x + 21\): \[ 4 \cdot -6x^2 = -24x^2 \] \[ 4 \cdot 8x = 32x \] \[ 4 \cdot 21 = 84 \]
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Multiply \(-2x\) by each term in the polynomial \(-6x^2 + 8x + 21\): \[ -2x \cdot -6x^2 = 12x^3 \] \[ -2x \cdot 8x = -16x^2 \] \[ -2x \cdot 21 = -42x \]
Now combine all the results together: \[ -24x^2 + 32x + 84 + 12x^3 - 16x^2 - 42x \]
Now we rearrange and group like terms: \[ 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84 \] \[ 12x^3 - 40x^2 - 10x + 84 \]
So the final expression after simplification is: \[ 12x^3 - 40x^2 - 10x + 84 \]
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