Solve (x-2y)(2x+3y+4)-(x+y)(3-2x-5y)

$$( x - 2 y ) ( 2 x + 3 y + 4 ) - ( x + y ) ( 3 - 2 x - 5 y )$$

Evaluate

$$4x^{2}+6xy+x-y^{2}-11y$$

Solution Steps

Apply the distributive property by multiplying each term of x-2y by each term of 2x+3y+4.

$$2x^{2}+3xy+4x-4yx-6y^{2}-8y-\left(x+y\right)\left(3-2x-5y\right)$$

Combine 3xy and -4yx to get -xy.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(x+y\right)\left(3-2x-5y\right)$$

Apply the distributive property by multiplying each term of x+y by each term of 3-2x-5y.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(3x-2x^{2}-5xy+3y-2yx-5y^{2}\right)$$

Combine -5xy and -2yx to get -7xy.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(3x-2x^{2}-7xy+3y-5y^{2}\right)$$

To find the opposite of 3x-2x^{2}-7xy+3y-5y^{2}, find the opposite of each term.

$$2x^{2}-xy+4x-6y^{2}-8y-3x-\left(-2x^{2}\right)-\left(-7xy\right)-3y-\left(-5y^{2}\right)$$

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

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}-\left(-7xy\right)-3y-\left(-5y^{2}\right)$$

The opposite of -7xy is 7xy.

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}+7xy-3y-\left(-5y^{2}\right)$$

The opposite of -5y^{2} is 5y^{2}.

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}+7xy-3y+5y^{2}$$

Combine 4x and -3x to get x.

$$2x^{2}-xy+x-6y^{2}-8y+2x^{2}+7xy-3y+5y^{2}$$

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

$$4x^{2}-xy+x-6y^{2}-8y+7xy-3y+5y^{2}$$

Combine -xy and 7xy to get 6xy.

$$4x^{2}+6xy+x-6y^{2}-8y-3y+5y^{2}$$

Combine -8y and -3y to get -11y.

$$4x^{2}+6xy+x-6y^{2}-11y+5y^{2}$$

Combine -6y^{2} and 5y^{2} to get -y^{2}.

$$4x^{2}+6xy+x-y^{2}-11y$$

Expand

$$4x^{2}+6xy+x-y^{2}-11y$$

Solution Steps

Apply the distributive property by multiplying each term of x-2y by each term of 2x+3y+4.

$$2x^{2}+3xy+4x-4yx-6y^{2}-8y-\left(x+y\right)\left(3-2x-5y\right)$$

Combine 3xy and -4yx to get -xy.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(x+y\right)\left(3-2x-5y\right)$$

Apply the distributive property by multiplying each term of x+y by each term of 3-2x-5y.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(3x-2x^{2}-5xy+3y-2yx-5y^{2}\right)$$

Combine -5xy and -2yx to get -7xy.

$$2x^{2}-xy+4x-6y^{2}-8y-\left(3x-2x^{2}-7xy+3y-5y^{2}\right)$$

To find the opposite of 3x-2x^{2}-7xy+3y-5y^{2}, find the opposite of each term.

$$2x^{2}-xy+4x-6y^{2}-8y-3x-\left(-2x^{2}\right)-\left(-7xy\right)-3y-\left(-5y^{2}\right)$$

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

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}-\left(-7xy\right)-3y-\left(-5y^{2}\right)$$

The opposite of -7xy is 7xy.

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}+7xy-3y-\left(-5y^{2}\right)$$

The opposite of -5y^{2} is 5y^{2}.

$$2x^{2}-xy+4x-6y^{2}-8y-3x+2x^{2}+7xy-3y+5y^{2}$$

Combine 4x and -3x to get x.

$$2x^{2}-xy+x-6y^{2}-8y+2x^{2}+7xy-3y+5y^{2}$$

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

$$4x^{2}-xy+x-6y^{2}-8y+7xy-3y+5y^{2}$$

Combine -xy and 7xy to get 6xy.

$$4x^{2}+6xy+x-6y^{2}-8y-3y+5y^{2}$$

Combine -8y and -3y to get -11y.

$$4x^{2}+6xy+x-6y^{2}-11y+5y^{2}$$

Combine -6y^{2} and 5y^{2} to get -y^{2}.

$$4x^{2}+6xy+x-y^{2}-11y$$