Question

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:
\[
\boxed{12x^3 - 40x^2 - 10x + 84}
\]