Perform the operation(s) and simplify completely. (4−2x)(−6x2+10x+214−2x) (1 point) Responses −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 −6x2+10x+21 negative 6 x squared plus 10 x plus 21 −x2−2x+12

Question

GPT-4o mini · Answer

To simplify the expression $(4−2x)(−6x^2+10x+21−2x)$, we first simplify the expression inside the parentheses: $$ -6x^2 + 10x + 21 - 2x = -6x^2 + 8x + 21 $$ So now we need to multiply $ (4 - 2x) $ by $ (-6x^2 + 8x + 21) $: $$ (4 - 2x)(-6x^2 + 8x + 21) $$ Distributing gives us: $$ 4 \cdot (-6x^2) + 4 \cdot 8x + 4 \cdot 21 - 2x \cdot (-6x^2) - 2x \cdot 8x - 2x \cdot 21 $$ Calculating each term: 1. $ 4 \cdot (-6x^2) = -24x^2 $ 2. $ 4 \cdot 8x = 32x $ 3. $ 4 \cdot 21 = 84 $ 4. $ -2x \cdot (-6x^2) = 12x^3 $ 5. $ -2x \cdot 8x = -16x^2 $ 6. $ -2x \cdot 21 = -42x $ Now putting these all together, we have: $$ 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84 $$ Combining like terms: - For $x^3$: $12x^3$ - For $x^2$: $-24x^2 - 16x^2 = -40x^2$ - For $x$: $32x - 42x = -10x$ - Constant: $84$ Putting it all together gives us: $$ 12x^3 - 40x^2 - 10x + 84 $$ Thus, the simplified expression is: $$ \boxed{12x^3 - 40x^2 - 10x + 84} $$ However, it looks like I may have misinterpreted the signs. The answer choices suggest a different simplification pathway may need to be checked. Let me correct that right away, looking closely at the choices; I might have made a mistake in my algebraic handling. Let's recalculate it more carefully. Calculating $ (4-2x)(-6x^2 + 8x + 21) $: 1. **Distributing $4$**: - $ 4 \cdot -6x^2 = -24x^2 $ - $ 4 \cdot 8x = 32x $ - $ 4 \cdot 21 = 84 $ 2. **Distributing $-2x$**: - $ -2x \cdot -6x^2 = 12x^3 $ - $ -2x \cdot 8x = -16x^2 $ - $ -2x \cdot 21 = -42x $ Combining all: The first part $ 12x^3 - 24x^2 - 16x^2 + 32x - 42x + 84$: Combine those respective parts: - The $x^2$ terms combine to $-40x^2$ - The $x$ terms combine to $-10x$ Thus correctly assembling: $$ \boxed{12x^3 - 40x^2 - 10x + 84} $$ I apologize for the confusion and appreciate your patience. This result reflects a detailed calculation based on initial problem setup and may need proper analysis of the responses provided. Each could potentially reflect parts of this calculation. Please review as appropriate according to test structure.