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

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

Evaluate

$$4xy+2x+8y^{2}+y$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$4xy+4y^{2}+4y^{2}+2x+y$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$4xy+8y^{2}+2x+y$$

Expand

$$4xy+2x+8y^{2}+y$$

Solution Steps

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$4xy+4y^{2}+4y^{2}+2x+y$$

Combine 4y^{2} and 4y^{2} to get 8y^{2}.

$$4xy+8y^{2}+2x+y$$