Question

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

Explanation:

Step1: Analyze the coefficient relationship

We know that when we multiply \(x\) by the term in the top - middle cell (let's call it \(ax\)) to get \(-x^{2}\). Using the rule of polynomial multiplication \((x)\times(ax)=ax^{2}\). Since \(ax^{2}=-x^{2}\), then \(a = - 1\).

Step2: Fill in the other cells

For the cell below \(-x^{2}\) (in the \(+2\) row and \(-x\) column): \((+2)\times(-x)=-2x\)
For the last cell (constant term):
The original polynomial is \(-x^{3}-3x^{2}+4\).
The sum of the products from the table: \((-x^{3}-x^{2}-2x^{2}-2x)+4\).
Combining like terms \(-x^{3}-3x^{2}-2x + 4\). The last cell (constant term) is \(4\) (because when we divide \(-x^{3}-3x^{2}+4\) by \(x + 2\), we can also think of it as \((x + 2)(-x^{2}-x)+4\))

Answer:

The missing value in the top - middle cell is \(-x\), the value in the \(+2\) row and \(-x\) column is \(-2x\), and the value in the last cell (constant term) is \(4\)