Perform the operation(s) and simplify completely.(4−2x)(−6x2+10x+214−2x) (1 point)Responses−x3+5x2−6x+3negative x cubed plus 5 x squared minus 6 x plus 3−x2−2x+12negative x squared minus 2 x plus 12−6x2+10x+21negative 6 x squared plus 10 x plus 21−x2+2x−8

Answer 1

To simplify the expression \((4 - 2x)(-6x^2 + 10x + 21 - 2x)\), we start by simplifying the expression inside the parentheses.

First, combine like terms in \(-6x^2 + 10x + 21 - 2x\): \[ -6x^2 + (10x - 2x) + 21 = -6x^2 + 8x + 21 \]

Now our expression is: \[ (4 - 2x)(-6x^2 + 8x + 21) \]

Next, we apply the distributive property (also known as the FOIL method for binomials). We will multiply \(4\) by each term in \(-6x^2 + 8x + 21\) and then do the same for \(-2x\).

1. Multiply \(4\) and \(-6x^2\): \[ 4 \cdot -6x^2 = -24x^2 \]

2. Multiply \(4\) and \(8x\): \[ 4 \cdot 8x = 32x \]

3. Multiply \(4\) and \(21\): \[ 4 \cdot 21 = 84 \]

4. Now, multiply \(-2x\) and \(-6x^2\): \[ -2x \cdot -6x^2 = 12x^3 \]

5. Multiply \(-2x\) and \(8x\): \[ -2x \cdot 8x = -16x^2 \]

6. Multiply \(-2x\) and \(21\): \[ -2x \cdot 21 = -42x \]

Now combine all the results: \[ 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84 \]

Combining like terms:

• The \(x^3\) term: \(12x^3\)
• The \(x^2\) terms: \(-24x^2 - 16x^2 = -40x^2\)
• The \(x\) terms: \(32x - 42x = -10x\)
• The constant term: \(84\)

Putting it all together, we get: \[ 12x^3 - 40x^2 - 10x + 84 \]

Thus, the fully simplified result is: \[ \boxed{12x^3 - 40x^2 - 10x + 84} \]