fill in the missing values below one at a time to find the quotient when 2x^3 - 17x^2 - 20x + 15 is divided…

Explanation:

Step1: Multiply $2x$ by $x^2$

We are using long - division in polynomial form. When we multiply the first term of the divisor $2x$ by the first term of the quotient $x^2$, we get $2x^3$. To find the second term in the second row, we multiply the second term of the divisor $2x$ by the second - degree term of the quotient (which we will find as we continue the division). First, we know that when dividing $2x^3-17x^2 - 20x + 15$ by $2x + 3$, we start with the leading terms. The leading term of the dividend is $2x^3$ and the leading term of the divisor is $2x$. $\frac{2x^3}{2x}=x^2$. Now, to find the missing value in the second row, we multiply the second term of the divisor $2x$ by the next term of the quotient. We first consider the subtraction of the product of the divisor and the quotient terms from the dividend.

We multiply $2x$ by the next term of the quotient (which we find by looking at the remaining part of the dividend after subtracting the product of the first - found quotient term and the divisor). The next step is to multiply $2x$ by the coefficient of the $x$ - term in the quotient.

We know that when we multiply the divisor $2x + 3$ by the quotient term by term. The second term in the second row is obtained by multiplying $2x$ by the $x$ - term of the quotient.

We perform polynomial long - division. The first step gives us the quotient term $x^2$. Now, we consider the $x^2$ term in the quotient and multiply the divisor's terms by it.

We multiply $2x$ by $x^2$ to get $2x^3$ and then we multiply $2x$ by the next term of the quotient. To find the missing value in the second row, we note that we are building the product of the divisor $2x+3$ and the quotient.

The missing value in the second row is obtained by multiplying $2x$ by the $x$ - term of the quotient.

We know that when we multiply $2x$ by $x^2$ we get $2x^3$. Now, we consider the $x$ - term in the quotient. Let the quotient be $x^2+ax + b$. When we multiply $(2x + 3)(x^2+ax + b)=2x^3+(2a)x^2+(2b + 3a)x+3b$.

We first focus on the $x^2$ terms. We know that $2x^3-17x^2-20x + 15=(2x + 3)(x^2+ax + b)$.

The missing value in the second row is obtained by multiplying $2x$ by the $x$ - term of the quotient. Since we are dividing $2x^3-17x^2-20x + 15$ by $2x + 3$, and the first term of the quotient is $x^2$.

We multiply $2x$ by the $x$ - term of the quotient. The $x$ - term of the quotient is found by considering the $x^2$ terms in the dividend and the product of the divisor and the quotient.

We know that when we multiply $2x$ by $x^2$ we get $2x^3$. Now, we consider the subtraction of the product of the divisor and the quotient from the dividend.

The missing value in the second row is $- 10x^2$. Because when we perform polynomial long - division of $2x^3-17x^2-20x + 15$ by $2x + 3$, the first term of the quotient is $x^2$. Then we multiply $2x$ by the $x$ - term of the quotient. The $x$ - term of the quotient is $- 10x$ (by comparing the $x^2$ terms in the dividend and the product of the divisor and the quotient). So, $2x\times(- 10x)=-20x^2$.

Step2: Multiply $3$ by $x^2$

We multiply the second term of the divisor $3$ by the first term of the quotient $x^2$ to get $3x^2$. This is already given in the table.

Answer:

The missing value in the box is $-20x^2$