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

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

Evaluate

$$-6xy$$

Solution Steps

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

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

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}.

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

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

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

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

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

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

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

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

$$-2xy-4xy$$

Combine -2xy and -4xy to get -6xy.

$$-6xy$$

Expand

$$-6xy$$

Solution Steps

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

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

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}.

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

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

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

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

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

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

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

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

$$-2xy-4xy$$

Combine -2xy and -4xy to get -6xy.

$$-6xy$$