Solve left(2x+yright)^2-(x-y)(x+y)-5(x-y)

$${ \left(2x+y \right) }^{ 2 } -(x-y)(x+y)-5(x-y)$$

Evaluate

$$\left(2x+y\right)^{2}+y^{2}-x^{2}+5y-5x$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+y\right)^{2}.

$$4x^{2}+4xy+y^{2}-\left(x-y\right)\left(x+y\right)-5\left(x-y\right)$$

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4x^{2}+4xy+y^{2}-\left(x^{2}-y^{2}\right)-5\left(x-y\right)$$

To find the opposite of x^{2}-y^{2}, find the opposite of each term.

$$4x^{2}+4xy+y^{2}-x^{2}+y^{2}-5\left(x-y\right)$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+4xy+y^{2}+y^{2}-5\left(x-y\right)$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$3x^{2}+4xy+2y^{2}-5\left(x-y\right)$$

Use the distributive property to multiply -5 by x-y.

$$3x^{2}+4xy+2y^{2}-5x+5y$$

Expand

$$3x^{2}+4xy-5x+2y^{2}+5y$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+y\right)^{2}.

$$4x^{2}+4xy+y^{2}-\left(x-y\right)\left(x+y\right)-5\left(x-y\right)$$

Consider \left(x-y\right)\left(x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$4x^{2}+4xy+y^{2}-\left(x^{2}-y^{2}\right)-5\left(x-y\right)$$

To find the opposite of x^{2}-y^{2}, find the opposite of each term.

$$4x^{2}+4xy+y^{2}-x^{2}+y^{2}-5\left(x-y\right)$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+4xy+y^{2}+y^{2}-5\left(x-y\right)$$

Combine y^{2} and y^{2} to get 2y^{2}.

$$3x^{2}+4xy+2y^{2}-5\left(x-y\right)$$

Use the distributive property to multiply -5 by x-y.

$$3x^{2}+4xy+2y^{2}-5x+5y$$