To determine the quotient when [tex]\( (x + 2) \)[/tex] is divided into the polynomial [tex]\( 2x^2 - 2x - 12 \)[/tex], we use the method of synthetic division. Here, we divide the polynomial [tex]\( 2x^2 - 2x - 12 \)[/tex] by [tex]\( (x + 2) \)[/tex].
### Synthetic Division Process
1. Set up the problem:
- Divisor: The root of [tex]\( (x + 2) \)[/tex] is [tex]\(-2\)[/tex].
- Coefficients of the polynomial: [2, -2, -12].
2. Write down the coefficients:
[tex]\[
2, -2, -12
\][/tex]
3. Carry down the leading coefficient:
- The first coefficient is 2.
4. Multiply and add:
- Multiply [tex]\(-2\)[/tex] (the root) by the leading coefficient (2):
[tex]\( -2 \times 2 = -4 \)[/tex]
- Add this product to the next coefficient (-2):
[tex]\( -2 + (-4) = -6 \)[/tex]
- Now, take [tex]\(-2\)[/tex] and multiply it by the updated coefficient (-6):
[tex]\( -2 \times -6 = 12 \)[/tex]
- Add this product to the next coefficient (-12):
[tex]\( -12 + 12 = 0 \)[/tex]
5. Write down the quotient and remainder:
- The quotient is the sequence of coefficients obtained from the synthetic division, excluding the last number (which is the remainder).
- Quotient: The coefficients are [tex]\( 2 \)[/tex] and [tex]\( -6 \)[/tex], so the polynomial is [tex]\( 2x - 6 \)[/tex].
### Result
The quotient when dividing [tex]\( (x + 2) \)[/tex] into [tex]\( 2x^2 - 2x - 12 \)[/tex] is [tex]\( 2x - 6 \)[/tex], and the remainder is 0. Therefore, the correct answer is:
C. [tex]\( 2x - 6 \)[/tex] with no remainder.
