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

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

Evaluate

$$y^{2}+8xy-2x^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.

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

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

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

To find the opposite of 4x^{2}-4xy+y^{2}, find the opposite of each term.

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

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

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

Combine 4xy and 4xy to get 8xy.

$$-2x^{2}+8xy+2y^{2}-y^{2}$$

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

$$-2x^{2}+8xy+y^{2}$$

Expand

$$y^{2}+8xy-2x^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by x^{2}+2xy+y^{2}.

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

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

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

To find the opposite of 4x^{2}-4xy+y^{2}, find the opposite of each term.

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

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

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

Combine 4xy and 4xy to get 8xy.

$$-2x^{2}+8xy+2y^{2}-y^{2}$$

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

$$-2x^{2}+8xy+y^{2}$$