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

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

Evaluate

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

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}.

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

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

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

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

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

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

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

Expand

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

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}.

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

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

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

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

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

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

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