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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

Multiply -2 and -1 to get 2.

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

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

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

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

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

Expand

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

Solution Steps

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

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

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

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

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

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

Multiply -2 and -1 to get 2.

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

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

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

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

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