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

Explanation:

Step1: Use polynomial long - division

We perform polynomial long - division of (2x^{3}-17x^{2}-20x + 15) by (2x+3). First, divide the leading term of the dividend (2x^{3}) by the leading term of the divisor (2x). (\frac{2x^{3}}{2x}=x^{2}).

Step2: Multiply and subtract

Multiply (2x + 3) by (x^{2}) to get (2x^{3}+3x^{2}). Subtract this from the dividend: ((2x^{3}-17x^{2}-20x + 15)-(2x^{3}+3x^{2})=-20x^{2}-20x + 15).

Step3: Divide the new leading term

Divide the leading term of (-20x^{2}-20x + 15) (which is (-20x^{2})) by the leading term of (2x+3) (which is (2x)). (\frac{-20x^{2}}{2x}=-10x).

Step4: Multiply and subtract again

Multiply (2x + 3) by (-10x) to get (-20x^{2}-30x). Subtract this from (-20x^{2}-20x + 15): ((-20x^{2}-20x + 15)-(-20x^{2}-30x)=10x + 15).

Step5: Divide the new leading term

Divide the leading term of (10x + 15) (which is (10x)) by the leading term of (2x+3) (which is (2x)). (\frac{10x}{2x}=5).

Step6: Multiply and subtract one last time

Multiply (2x + 3) by (5) to get (10x+15). Subtract this from (10x + 15): ((10x + 15)-(10x+15)=0).

Answer:

(x^{2}-10x + 5)