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

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

Evaluate

$$y\left(y-4\right)$$

Solution Steps

Use the distributive property to multiply 2 by 1-x+3y.

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

Use the distributive property to multiply 2-2x+6y by x-2y-1 and combine like terms.

$$4x-10y-2-2x^{2}+10xy-12y^{2}+\left(3y-x+1\right)^{2}+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Square 3y-x+1.

$$4x-10y-2-2x^{2}+10xy-12y^{2}+x^{2}-6xy-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

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

$$4x-10y-2-x^{2}+10xy-12y^{2}-6xy-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine 10xy and -6xy to get 4xy.

$$4x-10y-2-x^{2}+4xy-12y^{2}-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine 4x and -2x to get 2x.

$$2x-10y-2-x^{2}+4xy-12y^{2}+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine -12y^{2} and 9y^{2} to get -3y^{2}.

$$2x-10y-2-x^{2}+4xy-3y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine -10y and 6y to get -4y.

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

Add -2 and 1 to get -1.

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

Consider \left(x-2y-1\right)\left(x-2y+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}, where a=x-2y and b=1. Square 1.

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

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

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

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

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

Combine 4xy and -4xy to get 0.

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

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

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

Subtract 1 from -1 to get -2.

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

Use the distributive property to multiply -2 by x-1.

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

Combine 2x and -2x to get 0.

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

Add -2 and 2 to get 0.

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

Expand

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

Solution Steps

Use the distributive property to multiply 2 by 1-x+3y.

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

Use the distributive property to multiply 2-2x+6y by x-2y-1 and combine like terms.

$$4x-10y-2-2x^{2}+10xy-12y^{2}+\left(3y-x+1\right)^{2}+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Square 3y-x+1.

$$4x-10y-2-2x^{2}+10xy-12y^{2}+x^{2}-6xy-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

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

$$4x-10y-2-x^{2}+10xy-12y^{2}-6xy-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine 10xy and -6xy to get 4xy.

$$4x-10y-2-x^{2}+4xy-12y^{2}-2x+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine 4x and -2x to get 2x.

$$2x-10y-2-x^{2}+4xy-12y^{2}+9y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine -12y^{2} and 9y^{2} to get -3y^{2}.

$$2x-10y-2-x^{2}+4xy-3y^{2}+6y+1+\left(x-2y-1\right)\left(x-2y+1\right)-2\left(x-1\right)$$

Combine -10y and 6y to get -4y.

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

Add -2 and 1 to get -1.

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

Consider \left(x-2y-1\right)\left(x-2y+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}, where a=x-2y and b=1. Square 1.

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

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

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

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

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

Combine 4xy and -4xy to get 0.

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

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

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

Subtract 1 from -1 to get -2.

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

Use the distributive property to multiply -2 by x-1.

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

Combine 2x and -2x to get 0.

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

Add -2 and 2 to get 0.

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