Solve (2x+y)^{2}-(x-2y)^{2}-8xy+(x-y)^2+2(x+y)^2-3(x^2+y^2)

$$( 2 x + y ) ^ { 2 } - ( x - 2 y ) ^ { 2 } - 8 x y + ( x - y ) ^ { 2 } + 2 ( x + y ) ^ { 2 } - 3 ( x ^ { 2 } + y ^ { 2 } )$$

Evaluate

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

Solution Steps

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

$$4x^{2}+4xy+y^{2}-\left(x-2y\right)^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$4x^{2}+4xy+y^{2}-\left(x^{2}-4xy+4y^{2}\right)-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$4x^{2}+4xy+y^{2}-x^{2}+4xy-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$3x^{2}+4xy+y^{2}+4xy-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

Combine 4xy and 4xy to get 8xy.

$$3x^{2}+8xy+y^{2}-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$3x^{2}+8xy-3y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

Combine 8xy and -8xy to get 0.

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

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

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

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

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

Combine -3y^{2} and y^{2} to get -2y^{2}.

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

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

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

Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.

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

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

$$6x^{2}-2y^{2}-2xy+4xy+2y^{2}-3\left(x^{2}+y^{2}\right)$$

Combine -2xy and 4xy to get 2xy.

$$6x^{2}-2y^{2}+2xy+2y^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$6x^{2}+2xy-3\left(x^{2}+y^{2}\right)$$

Use the distributive property to multiply -3 by x^{2}+y^{2}.

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

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

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

Expand

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

Solution Steps

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

$$4x^{2}+4xy+y^{2}-\left(x-2y\right)^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$4x^{2}+4xy+y^{2}-\left(x^{2}-4xy+4y^{2}\right)-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$4x^{2}+4xy+y^{2}-x^{2}+4xy-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$3x^{2}+4xy+y^{2}+4xy-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

Combine 4xy and 4xy to get 8xy.

$$3x^{2}+8xy+y^{2}-4y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$3x^{2}+8xy-3y^{2}-8xy+\left(x-y\right)^{2}+2\left(x+y\right)^{2}-3\left(x^{2}+y^{2}\right)$$

Combine 8xy and -8xy to get 0.

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

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

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

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

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

Combine -3y^{2} and y^{2} to get -2y^{2}.

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

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

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

Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.

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

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

$$6x^{2}-2y^{2}-2xy+4xy+2y^{2}-3\left(x^{2}+y^{2}\right)$$

Combine -2xy and 4xy to get 2xy.

$$6x^{2}-2y^{2}+2xy+2y^{2}-3\left(x^{2}+y^{2}\right)$$

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

$$6x^{2}+2xy-3\left(x^{2}+y^{2}\right)$$

Use the distributive property to multiply -3 by x^{2}+y^{2}.

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

Combine 6x^{2} and -3x^{2} to get 3x^{2}.

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