Solve (2x-y)^2-(2x-y)(2x+y)+(3y)+(2y)^2

$$( 2 x - y ) ^ { 2 } - ( 2 x - y ) ( 2 x + y ) + ( 3 y ) + ( 2 y ) ^ { 2 }$$

Evaluate

$$y\left(3+6y-4x\right)$$

Solution Steps

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

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

Consider \left(2x-y\right)\left(2x+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}.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of 4x^{2}-y^{2}, find the opposite of each term.

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

Combine 4x^{2} and -4x^{2} to get 0.

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

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

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

Expand \left(2y\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Combine 2y^{2} and 4y^{2} to get 6y^{2}.

$$-4xy+6y^{2}+3y$$

Expand

$$3y+6y^{2}-4xy$$

Solution Steps

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

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

Consider \left(2x-y\right)\left(2x+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}.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

To find the opposite of 4x^{2}-y^{2}, find the opposite of each term.

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

Combine 4x^{2} and -4x^{2} to get 0.

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

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

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

Expand \left(2y\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Combine 2y^{2} and 4y^{2} to get 6y^{2}.

$$-4xy+6y^{2}+3y$$