Solve (x+2y)(2y+x)+(-x-2y)^2+(2x-y)(-2x+y)+2x(x-6y)

$$( x + 2 y ) ( 2 y + x ) + ( - x - 2 y ) ^ { 2 } + ( 2 x - y ) ( - 2 x + y ) + 2 x ( x - 6 y )$$

Evaluate

$$7y^{2}$$

Solution Steps

Multiply x+2y and 2y+x to get \left(x+2y\right)^{2}.

$$\left(x+2y\right)^{2}+\left(-x-2y\right)^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$x^{2}+4xy+4y^{2}+\left(-x-2y\right)^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$x^{2}+4xy+4y^{2}+\left(-x\right)^{2}-4\left(-x\right)y+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}+4xy+4y^{2}+x^{2}-4\left(-x\right)y+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Multiply -4 and -1 to get 4.

$$x^{2}+4xy+4y^{2}+x^{2}+4xy+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$2x^{2}+4xy+4y^{2}+4xy+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Combine 4xy and 4xy to get 8xy.

$$2x^{2}+8xy+4y^{2}+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$2x^{2}+8xy+8y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Use the distributive property to multiply 2x-y by -2x+y and combine like terms.

$$2x^{2}+8xy+8y^{2}-4x^{2}+4xy-y^{2}+2x\left(x-6y\right)$$

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

$$-2x^{2}+8xy+8y^{2}+4xy-y^{2}+2x\left(x-6y\right)$$

Combine 8xy and 4xy to get 12xy.

$$-2x^{2}+12xy+8y^{2}-y^{2}+2x\left(x-6y\right)$$

Combine 8y^{2} and -y^{2} to get 7y^{2}.

$$-2x^{2}+12xy+7y^{2}+2x\left(x-6y\right)$$

Use the distributive property to multiply 2x by x-6y.

$$-2x^{2}+12xy+7y^{2}+2x^{2}-12xy$$

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

$$12xy+7y^{2}-12xy$$

Combine 12xy and -12xy to get 0.

$$7y^{2}$$

Expand

$$7y^{2}$$

Solution Steps

Multiply x+2y and 2y+x to get \left(x+2y\right)^{2}.

$$\left(x+2y\right)^{2}+\left(-x-2y\right)^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$x^{2}+4xy+4y^{2}+\left(-x-2y\right)^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$x^{2}+4xy+4y^{2}+\left(-x\right)^{2}-4\left(-x\right)y+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}+4xy+4y^{2}+x^{2}-4\left(-x\right)y+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Multiply -4 and -1 to get 4.

$$x^{2}+4xy+4y^{2}+x^{2}+4xy+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$2x^{2}+4xy+4y^{2}+4xy+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Combine 4xy and 4xy to get 8xy.

$$2x^{2}+8xy+4y^{2}+4y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

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

$$2x^{2}+8xy+8y^{2}+\left(2x-y\right)\left(-2x+y\right)+2x\left(x-6y\right)$$

Use the distributive property to multiply 2x-y by -2x+y and combine like terms.

$$2x^{2}+8xy+8y^{2}-4x^{2}+4xy-y^{2}+2x\left(x-6y\right)$$

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

$$-2x^{2}+8xy+8y^{2}+4xy-y^{2}+2x\left(x-6y\right)$$

Combine 8xy and 4xy to get 12xy.

$$-2x^{2}+12xy+8y^{2}-y^{2}+2x\left(x-6y\right)$$

Combine 8y^{2} and -y^{2} to get 7y^{2}.

$$-2x^{2}+12xy+7y^{2}+2x\left(x-6y\right)$$

Use the distributive property to multiply 2x by x-6y.

$$-2x^{2}+12xy+7y^{2}+2x^{2}-12xy$$

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

$$12xy+7y^{2}-12xy$$

Combine 12xy and -12xy to get 0.

$$7y^{2}$$