fill in the missing values below one at a time to find the quotient when -2x^3 + 3x^2 + 4 is divided by x - 2.

Explanation:

Step1: Use polynomial long - division or synthetic division.

We use synthetic division. The divisor is (x - 2), so the value for synthetic division is (c = 2). The polynomial is (-2x^{3}+3x^{2}+0x + 4) (we write (0x) for the missing (x) - term).

Step2: Set up synthetic division.

Bring down the first coefficient (-2). Multiply (2\times(-2)=-4), add to the next coefficient: (3+( - 4)=-1). Then (2\times(-1)=-2), add to the next coefficient (0+( - 2)=-2). Then (2\times(-2)=-4), add to the last coefficient (4+( - 4)=0). The synthetic - division setup:

(2)	(-2)	(3)	(0)	(4)
		(-4)	(-2)	(-4)
	(-2)	(-1)	(-2)	(0)
The quotient is (-2x^{2}-x - 2).
We can also think of it in terms of the area - model (box method).
The first term of the quotient when dividing (-2x^{3}) by (x) is (-2x^{2}). Multiply ((x - 2)\times(-2x^{2})=-2x^{3}+4x^{2}). Subtract from the original polynomial: ((-2x^{3}+3x^{2}+4)-(-2x^{3}+4x^{2})=-x^{2}+4).
The next term of the quotient when dividing (-x^{2}) by (x) is (-x). Multiply ((x - 2)\times(-x)=-x^{2}+2x). Subtract: ((-x^{2}+4)-(-x^{2}+2x)=-2x + 4).
The last term of the quotient when dividing (-2x) by (x) is (-2). Multiply ((x - 2)\times(-2)=-2x + 4). Subtract: ((-2x + 4)-(-2x + 4)=0).
The box - method:
	(x)	(-2)
---	---	---
(-2x^{2})	(-2x^{3})	(4x^{2})
(-x)	(-x^{2})	(2x)
(-2)	(-2x)	(4)
The quotient is (-2x^{2}-x - 2).

Answer:

The quotient is (-2x^{2}-x - 2)