Solve (x-y-2)^2-(2x+y-1)^2-3(1-x)(x+2y+1)

$$( x - y - 2 ) ^ { 2 } - ( 2 x + y - 1 ) ^ { 2 } - 3 ( 1 - x ) ( x + 2 y + 1 )$$

Evaluate

$$0$$

Solution Steps

Square x-y-2.

$$x^{2}-2xy-4x+y^{2}+4y+4-\left(2x+y-1\right)^{2}-3\left(1-x\right)\left(x+2y+1\right)$$

Square 2x+y-1.

$$x^{2}-2xy-4x+y^{2}+4y+4-\left(4x^{2}+4xy-4x+y^{2}-2y+1\right)-3\left(1-x\right)\left(x+2y+1\right)$$

To find the opposite of 4x^{2}+4xy-4x+y^{2}-2y+1, find the opposite of each term.

$$x^{2}-2xy-4x+y^{2}+4y+4-4x^{2}-4xy+4x-y^{2}+2y-1-3\left(1-x\right)\left(x+2y+1\right)$$

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

$$-3x^{2}-2xy-4x+y^{2}+4y+4-4xy+4x-y^{2}+2y-1-3\left(1-x\right)\left(x+2y+1\right)$$

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

$$-3x^{2}-6xy-4x+y^{2}+4y+4+4x-y^{2}+2y-1-3\left(1-x\right)\left(x+2y+1\right)$$

Combine -4x and 4x to get 0.

$$-3x^{2}-6xy+y^{2}+4y+4-y^{2}+2y-1-3\left(1-x\right)\left(x+2y+1\right)$$

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

$$-3x^{2}-6xy+4y+4+2y-1-3\left(1-x\right)\left(x+2y+1\right)$$

Combine 4y and 2y to get 6y.

$$-3x^{2}-6xy+6y+4-1-3\left(1-x\right)\left(x+2y+1\right)$$

Subtract 1 from 4 to get 3.

$$-3x^{2}-6xy+6y+3-3\left(1-x\right)\left(x+2y+1\right)$$

Use the distributive property to multiply -3 by 1-x.

$$-3x^{2}-6xy+6y+3+\left(-3+3x\right)\left(x+2y+1\right)$$

Use the distributive property to multiply -3+3x by x+2y+1 and combine like terms.

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

Combine 6y and -6y to get 0.

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

Subtract 3 from 3 to get 0.

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

Combine -3x^{2} and 3x^{2} to get 0.

$$-6xy+6xy$$

Combine -6xy and 6xy to get 0.

$$0$$

Factor

$$0$$