Solve -2(4x-y)^2+5(-2x+y)^2

$$- 2 ( 4 x - y ) ^ { 2 } + 5 ( - 2 x + y ) ^ { 2 }$$

Evaluate

$$3y^{2}-4xy-12x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-y\right)^{2}.

$$-2\left(16x^{2}-8xy+y^{2}\right)+5\left(-2x+y\right)^{2}$$

Use the distributive property to multiply -2 by 16x^{2}-8xy+y^{2}.

$$-32x^{2}+16xy-2y^{2}+5\left(-2x+y\right)^{2}$$

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

$$-32x^{2}+16xy-2y^{2}+5\left(4x^{2}-4xy+y^{2}\right)$$

Use the distributive property to multiply 5 by 4x^{2}-4xy+y^{2}.

$$-32x^{2}+16xy-2y^{2}+20x^{2}-20xy+5y^{2}$$

Combine -32x^{2} and 20x^{2} to get -12x^{2}.

$$-12x^{2}+16xy-2y^{2}-20xy+5y^{2}$$

Combine 16xy and -20xy to get -4xy.

$$-12x^{2}-4xy-2y^{2}+5y^{2}$$

Combine -2y^{2} and 5y^{2} to get 3y^{2}.

$$-12x^{2}-4xy+3y^{2}$$

Expand

$$3y^{2}-4xy-12x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-y\right)^{2}.

$$-2\left(16x^{2}-8xy+y^{2}\right)+5\left(-2x+y\right)^{2}$$

Use the distributive property to multiply -2 by 16x^{2}-8xy+y^{2}.

$$-32x^{2}+16xy-2y^{2}+5\left(-2x+y\right)^{2}$$

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

$$-32x^{2}+16xy-2y^{2}+5\left(4x^{2}-4xy+y^{2}\right)$$

Use the distributive property to multiply 5 by 4x^{2}-4xy+y^{2}.

$$-32x^{2}+16xy-2y^{2}+20x^{2}-20xy+5y^{2}$$

Combine -32x^{2} and 20x^{2} to get -12x^{2}.

$$-12x^{2}+16xy-2y^{2}-20xy+5y^{2}$$

Combine 16xy and -20xy to get -4xy.

$$-12x^{2}-4xy-2y^{2}+5y^{2}$$

Combine -2y^{2} and 5y^{2} to get 3y^{2}.

$$-12x^{2}-4xy+3y^{2}$$