Solve {l}{(x+y)^2+(x-y)(x+y)}{-5x(x-y)}

$$\left. \begin{array} { l } { ( x + y ) ^ { 2 } + ( x - y ) ( x + y ) } \\ { - 5 x ( x - y ) } \end{array} \right.$$

Evaluate

$$x\left(7y-3x\right)$$

Solution Steps

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

$$x^{2}+2xy+y^{2}+\left(x-y\right)\left(x+y\right)-5x\left(x-y\right)$$

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}.

$$x^{2}+2xy+y^{2}+x^{2}-y^{2}-5x\left(x-y\right)$$

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

$$2x^{2}+2xy+y^{2}-y^{2}-5x\left(x-y\right)$$

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

$$2x^{2}+2xy-5x\left(x-y\right)$$

Use the distributive property to multiply -5x by x-y.

$$2x^{2}+2xy-5x^{2}+5xy$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}+2xy+5xy$$

Combine 2xy and 5xy to get 7xy.

$$-3x^{2}+7xy$$

Expand

$$7xy-3x^{2}$$

Solution Steps

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

$$x^{2}+2xy+y^{2}+\left(x-y\right)\left(x+y\right)-5x\left(x-y\right)$$

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}.

$$x^{2}+2xy+y^{2}+x^{2}-y^{2}-5x\left(x-y\right)$$

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

$$2x^{2}+2xy+y^{2}-y^{2}-5x\left(x-y\right)$$

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

$$2x^{2}+2xy-5x\left(x-y\right)$$

Use the distributive property to multiply -5x by x-y.

$$2x^{2}+2xy-5x^{2}+5xy$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}+2xy+5xy$$

Combine 2xy and 5xy to get 7xy.

$$-3x^{2}+7xy$$