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

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

Evaluate

$$4x^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x+y.

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

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

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

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

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

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

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

$$4x^{2}$$

Expand

$$4x^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x+y.

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

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

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

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

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

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

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

$$4x^{2}$$