fill in the missing values below one at a time to find the quotient when -x^3 + 23x + 28 is divided by x + 4.

Explanation:

Step1: Determine the first - term of the quotient

We are dividing (-x^{3}+23x + 28) by (x + 4). To get the first - term of the quotient, divide the leading term of the dividend (-x^{3}) by the leading term of the divisor (x). So, (\frac{-x^{3}}{x}=-x^{2}).

Step2: Multiply the divisor by the first - term of the quotient

Multiply (x + 4) by (-x^{2}). We get (-x^{2}(x + 4)=-x^{3}-4x^{2}).

Step3: Determine the second - term of the quotient

The second column in the dividend has a term related to (x). We know that the sum of the terms in the second column gives the (x) - terms in the dividend. The (x) - terms in the dividend is (23x), and one of the terms in the second column is (16x). Let the second - term of the quotient be (ax). Then ((x + 4)\times ax=ax^{2}+4ax). Since we already have (-4x^{2}) in the second column from the previous step, and we know that the sum of the terms in the second column should give the (x) - terms in the dividend. We consider the (x) - terms. We know that (4ax=16x), so (a = 4). The second - term of the quotient is (4x).

Step4: Multiply the divisor by the second - term of the quotient

Multiply (x + 4) by (4x), we get (4x(x + 4)=4x^{2}+16x).

The missing values from left - to - right, top - to - bottom are (-x^{2}), (4x), (16x).

Answer:

The missing values are (-x^{2}), (4x), (16x)