Chapter 4.6, Problem 8WE

To determine

To Multiply: the given expression

Answer to Problem 8WE

The expression we get after multiplying using vertical form is $\boxed{4x^3-5x{y}^2-y^3-4x^{2}y}$

Explanation of Solution

Given information: $\begin{array}{l}2x^2-3xy-y^2\\ \underset{\_}{2x+y}\end{array}$

Concept Involved:

Multiply the second term of the bottom row with first row to get the 1^st partial product, then multiply the first term of bottom row with first row to get 2^nd partial product, then add the results of 1^st partial product and 2^nd partial product to get the final answer.

Calculation:

Multiplying y with first row to get the 1^st partial product

  $\begin{array}{l}2x^2-3xy-y^2\\ \underset{\_}{2x+y}\\ 2x^{2}y-3x{y}^2-y^3\to 1^{st}partialproduct\end{array}$

Multiplying 2x with the second row to the 2^nd partial product

  $\begin{array}{l}2x^2-3xy-y^2\\ \underset{\_}{2x+y}\\ 2x^{2}y-3x{y}^2-y^3\to 1^{st}partialproduct\\ 4x^3-6x^{2}y-2x{y}^2\to 2^{nd}partialproduct\end{array}$

Add the 1^st and 2^nd partial product to get the final answer

  $\begin{array}{l}2x^2-3xy-y^2\\ \underset{\_}{2x+y}\\ 2x^{2}y-3x{y}^2-y^3\to 1^{st}partialproduct\\ \underset{\_}{4x^3-6x^{2}y-2x{y}^2\to 2^{nd}partialproduct}\\ 4x^3-4x^{2}y-5x{y}^2-y^3\end{array}$

Conclusion:

The expression we get after multiplying using vertical form is $\boxed{4x^3-4x^{2}y-5x{y}^2-y^3}$