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

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

Evaluate

$$4x^{2}-12xy+10y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 4x^{2}-4xy+y^{2}.

$$8x^{2}-8xy+2y^{2}-4\left(x-y\right)\left(x+2y\right)$$

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

$$8x^{2}-8xy+2y^{2}+\left(-4x+4y\right)\left(x+2y\right)$$

Use the distributive property to multiply -4x+4y by x+2y and combine like terms.

$$8x^{2}-8xy+2y^{2}-4x^{2}-4xy+8y^{2}$$

Combine 8x^{2} and -4x^{2} to get 4x^{2}.

$$4x^{2}-8xy+2y^{2}-4xy+8y^{2}$$

Combine -8xy and -4xy to get -12xy.

$$4x^{2}-12xy+2y^{2}+8y^{2}$$

Combine 2y^{2} and 8y^{2} to get 10y^{2}.

$$4x^{2}-12xy+10y^{2}$$

Expand

$$4x^{2}-12xy+10y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 4x^{2}-4xy+y^{2}.

$$8x^{2}-8xy+2y^{2}-4\left(x-y\right)\left(x+2y\right)$$

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

$$8x^{2}-8xy+2y^{2}+\left(-4x+4y\right)\left(x+2y\right)$$

Use the distributive property to multiply -4x+4y by x+2y and combine like terms.

$$8x^{2}-8xy+2y^{2}-4x^{2}-4xy+8y^{2}$$

Combine 8x^{2} and -4x^{2} to get 4x^{2}.

$$4x^{2}-8xy+2y^{2}-4xy+8y^{2}$$

Combine -8xy and -4xy to get -12xy.

$$4x^{2}-12xy+2y^{2}+8y^{2}$$

Combine 2y^{2} and 8y^{2} to get 10y^{2}.

$$4x^{2}-12xy+10y^{2}$$