What is teh qquotient when -2x3 + 4x2 + 6 is divided by  2x3 + 4x2 + 6?A.x-1B.-xC.-1D.1

1. Break Down the Problem

We need to perform polynomial division of the polynomial $-2x^3 + 4x^2 + 6$ by the polynomial $2x^3 + 4x^2 + 6$. We will find the quotient of this division.

2. Relevant Concepts

When dividing polynomials, we can use long division or synthetic division. The goal is to find:

$\text{Quotient} = \frac{-2x^3 + 4x^2 + 6}{2x^3 + 4x^2 + 6}$

3. Analysis and Detail

Let's divide the polynomials:

1. Divide the leading terms: $\frac{-2x^3}{2x^3} = -1$

2. Multiply the entire divisor by -1: $-1 \cdot (2x^3 + 4x^2 + 6) = -2x^3 - 4x^2 - 6$

3. Subtract this result from the original polynomial: $(-2x^3 + 4x^2 + 6) - (-2x^3 - 4x^2 - 6) = (4x^2 + 4x^2) + (6 + 6) = 8x^2 + 12$

4. Remainder: Since $8x^2 + 12$ cannot be divided further by the divisor $2x^3 + 4x^2 + 6$, we stop here.

The quotient obtained from this division is:

$\text{Quotient} = -1$

4. Verify and Summarize

We have performed the division correctly and found that the quotient is $-1$. The remainder is $8x^2 + 12$, but it does not affect the quotient because it doesn't alter the integer nature of the quotient.

Final Answer

$-1$