Solve left(x+yright)left(x+yright)+left(x-yright)left(x-yright)+2left(x+yright)left(x-yright)

$$\left( x+y \right) \left( x+y \right) + \left( x-y \right) \left( x-y \right) +2 \left( x+y \right) \left( x-y \right)$$

Evaluate

$$4x^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

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

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

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

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

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

Apply the distributive property by multiplying each term of 2x+2y by each term of x-y.

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

Combine -2xy and 2yx to get 0.

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

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

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

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

$$4x^{2}$$

Expand

$$4x^{2}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Combine 2xy and -2xy to get 0.

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

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

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

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

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

Apply the distributive property by multiplying each term of 2x+2y by each term of x-y.

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

Combine -2xy and 2yx to get 0.

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

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

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

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

$$4x^{2}$$