Solve 5(x^2-2xy)-(x+y)^2-3(x+y)(x-y)+5

$$5 ( x ^ { 2 } - 2 x y ) - ( x + y ) ^ { 2 } - 3 ( x + y ) ( x - y ) + 5$$

Evaluate

$$x^{2}-12xy+2y^{2}+5$$

Solution Steps

Use the distributive property to multiply 5 by x^{2}-2xy.

$$5x^{2}-10xy-\left(x+y\right)^{2}-3\left(x+y\right)\left(x-y\right)+5$$

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

$$5x^{2}-10xy-\left(x^{2}+2xy+y^{2}\right)-3\left(x+y\right)\left(x-y\right)+5$$

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

$$5x^{2}-10xy-x^{2}-2xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-10xy-2xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Combine -10xy and -2xy to get -12xy.

$$4x^{2}-12xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Use the distributive property to multiply -3 by x+y.

$$4x^{2}-12xy-y^{2}+\left(-3x-3y\right)\left(x-y\right)+5$$

Use the distributive property to multiply -3x-3y by x-y and combine like terms.

$$4x^{2}-12xy-y^{2}-3x^{2}+3y^{2}+5$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}-12xy-y^{2}+3y^{2}+5$$

Combine -y^{2} and 3y^{2} to get 2y^{2}.

$$x^{2}-12xy+2y^{2}+5$$

Expand

$$x^{2}-12xy+2y^{2}+5$$

Solution Steps

Use the distributive property to multiply 5 by x^{2}-2xy.

$$5x^{2}-10xy-\left(x+y\right)^{2}-3\left(x+y\right)\left(x-y\right)+5$$

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

$$5x^{2}-10xy-\left(x^{2}+2xy+y^{2}\right)-3\left(x+y\right)\left(x-y\right)+5$$

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

$$5x^{2}-10xy-x^{2}-2xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-10xy-2xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Combine -10xy and -2xy to get -12xy.

$$4x^{2}-12xy-y^{2}-3\left(x+y\right)\left(x-y\right)+5$$

Use the distributive property to multiply -3 by x+y.

$$4x^{2}-12xy-y^{2}+\left(-3x-3y\right)\left(x-y\right)+5$$

Use the distributive property to multiply -3x-3y by x-y and combine like terms.

$$4x^{2}-12xy-y^{2}-3x^{2}+3y^{2}+5$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}-12xy-y^{2}+3y^{2}+5$$

Combine -y^{2} and 3y^{2} to get 2y^{2}.

$$x^{2}-12xy+2y^{2}+5$$