Solve (3x+2y-1)(3x+y)-(x+2y)(x-y)-8x(y+2x)

$$( 3 x + 2 y - 1 ) ( 3 x + y ) - ( x + 2 y ) ( x - y ) - 8 x ( y + 2 x )$$

Evaluate

$$-8x^{2}-3x+4y^{2}-y$$

Solution Steps

Apply the distributive property by multiplying each term of 3x+2y-1 by each term of 3x+y.

$$9x^{2}+3xy+6yx+2y^{2}-3x-y-\left(x+2y\right)\left(x-y\right)-8x\left(y+2x\right)$$

Combine 3xy and 6yx to get 9xy.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x+2y\right)\left(x-y\right)-8x\left(y+2x\right)$$

Apply the distributive property by multiplying each term of x+2y by each term of x-y.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x^{2}-xy+2yx-2y^{2}\right)-8x\left(y+2x\right)$$

Combine -xy and 2yx to get xy.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x^{2}+xy-2y^{2}\right)-8x\left(y+2x\right)$$

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

$$9x^{2}+9xy+2y^{2}-3x-y-x^{2}-xy-\left(-2y^{2}\right)-8x\left(y+2x\right)$$

The opposite of -2y^{2} is 2y^{2}.

$$9x^{2}+9xy+2y^{2}-3x-y-x^{2}-xy+2y^{2}-8x\left(y+2x\right)$$

Combine 9x^{2} and -x^{2} to get 8x^{2}.

$$8x^{2}+9xy+2y^{2}-3x-y-xy+2y^{2}-8x\left(y+2x\right)$$

Combine 9xy and -xy to get 8xy.

$$8x^{2}+8xy+2y^{2}-3x-y+2y^{2}-8x\left(y+2x\right)$$

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

$$8x^{2}+8xy+4y^{2}-3x-y-8x\left(y+2x\right)$$

Use the distributive property to multiply -8x by y+2x.

$$8x^{2}+8xy+4y^{2}-3x-y-8xy-16x^{2}$$

Combine 8xy and -8xy to get 0.

$$8x^{2}+4y^{2}-3x-y-16x^{2}$$

Combine 8x^{2} and -16x^{2} to get -8x^{2}.

$$-8x^{2}+4y^{2}-3x-y$$

Expand

$$-8x^{2}-3x+4y^{2}-y$$

Solution Steps

Apply the distributive property by multiplying each term of 3x+2y-1 by each term of 3x+y.

$$9x^{2}+3xy+6yx+2y^{2}-3x-y-\left(x+2y\right)\left(x-y\right)-8x\left(y+2x\right)$$

Combine 3xy and 6yx to get 9xy.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x+2y\right)\left(x-y\right)-8x\left(y+2x\right)$$

Apply the distributive property by multiplying each term of x+2y by each term of x-y.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x^{2}-xy+2yx-2y^{2}\right)-8x\left(y+2x\right)$$

Combine -xy and 2yx to get xy.

$$9x^{2}+9xy+2y^{2}-3x-y-\left(x^{2}+xy-2y^{2}\right)-8x\left(y+2x\right)$$

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

$$9x^{2}+9xy+2y^{2}-3x-y-x^{2}-xy-\left(-2y^{2}\right)-8x\left(y+2x\right)$$

The opposite of -2y^{2} is 2y^{2}.

$$9x^{2}+9xy+2y^{2}-3x-y-x^{2}-xy+2y^{2}-8x\left(y+2x\right)$$

Combine 9x^{2} and -x^{2} to get 8x^{2}.

$$8x^{2}+9xy+2y^{2}-3x-y-xy+2y^{2}-8x\left(y+2x\right)$$

Combine 9xy and -xy to get 8xy.

$$8x^{2}+8xy+2y^{2}-3x-y+2y^{2}-8x\left(y+2x\right)$$

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

$$8x^{2}+8xy+4y^{2}-3x-y-8x\left(y+2x\right)$$

Use the distributive property to multiply -8x by y+2x.

$$8x^{2}+8xy+4y^{2}-3x-y-8xy-16x^{2}$$

Combine 8xy and -8xy to get 0.

$$8x^{2}+4y^{2}-3x-y-16x^{2}$$

Combine 8x^{2} and -16x^{2} to get -8x^{2}.

$$-8x^{2}+4y^{2}-3x-y$$