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

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

Evaluate

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

Solution Steps

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

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

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

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

To find the opposite of 3x^{2}+2xy-y^{2}, find the opposite of each term.

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

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

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

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

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

Combine 4y^{2} and y^{2} to get 5y^{2}.

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

Combine 5y^{2} and -5y^{2} to get 0.

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

Expand

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

Solution Steps

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

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

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

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

To find the opposite of 3x^{2}+2xy-y^{2}, find the opposite of each term.

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

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

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

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

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

Combine 4y^{2} and y^{2} to get 5y^{2}.

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

Combine 5y^{2} and -5y^{2} to get 0.

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