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

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

Evaluate

$$3y\left(y-2x\right)$$

Solution Steps

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

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

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

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

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

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

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

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

Combine -2xy and -4xy to get -6xy.

$$-6xy-y^{2}+4y^{2}$$

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

$$-6xy+3y^{2}$$

Expand

$$3y^{2}-6xy$$

Solution Steps

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

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

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

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

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

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

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

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

Combine -2xy and -4xy to get -6xy.

$$-6xy-y^{2}+4y^{2}$$

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

$$-6xy+3y^{2}$$