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

Answer 1

To simplify the expression \( (4 - 2x)(-6x^2 + 10x + 21 - 2x) \), we first simplify the second expression inside the parentheses:

\[ -6x^2 + 10x + 21 - 2x = -6x^2 + (10x - 2x) + 21 = -6x^2 + 8x + 21 \]

Now, we rewrite the entire expression:

\[ (4 - 2x)(-6x^2 + 8x + 21) \]

Next, we will use the distributive property (often called the FOIL method for binomials) to multiply these two expressions. We can distribute \( 4 \) and \( -2x \) across the polynomial:

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 we combine all of these results:

\[ 12x^3 + (-24x^2 - 16x^2) + (32x - 42x) + 84 \]

This simplifies to:

\[ 12x^3 - 40x^2 - 10x + 84 \]

Now we can write the expression in standard form:

\[ 12x^3 - 40x^2 - 10x + 84 \]

Thus, our simplified polynomial is:

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

This expression can be irreducible in its current form, so we leave it as is.