Solve left(x+yright)^2-4x(x+y)-2y(x+y)

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

Evaluate

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

Solution Steps

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

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

Use the distributive property to multiply -4x by x+y.

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

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

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

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

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

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

$$-3x^{2}-2xy+y^{2}-2yx-2y^{2}$$

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

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

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

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

Expand

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

Solution Steps

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

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

Use the distributive property to multiply -4x by x+y.

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

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

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

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

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

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

$$-3x^{2}-2xy+y^{2}-2yx-2y^{2}$$

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

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

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

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