Solve (x-y-4y^2/x-y)(x+y-4x^2/x+y)div{3(x+y)-8xy/x-y}

$$( x - y - \frac { 4 y ^ { 2 } } { x - y } ) ( x + y - \frac { 4 x ^ { 2 } } { x + y } ) \div \{ 3 ( x + y ) - \frac { 8 x y } { x - y } \}$$

Evaluate

$$y-x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x-y times \frac{x-y}{x-y}.

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

Since \frac{\left(x-y\right)\left(x-y\right)}{x-y} and \frac{4y^{2}}{x-y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-y\right)\left(x-y\right)-4y^{2}.

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

Combine like terms in x^{2}-xy-xy+y^{2}-4y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x+y times \frac{x+y}{x+y}.

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

Since \frac{\left(x+y\right)\left(x+y\right)}{x+y} and \frac{4x^{2}}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+y\right)\left(x+y\right)-4x^{2}.

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

Combine like terms in x^{2}+xy+xy+y^{2}-4x^{2}.

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

Multiply \frac{x^{2}-3y^{2}-2xy}{x-y} times \frac{-3x^{2}+y^{2}+2xy}{x+y} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{\left(x^{2}-3y^{2}-2xy\right)\left(-3x^{2}+y^{2}+2xy\right)}{\left(x-y\right)\left(x+y\right)}.

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

Extract the negative sign in -x+y.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

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

Expand the expression.

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

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

$$\frac{-3x^{2}+8xy+3y^{2}}{3x+3y-\frac{8xy}{x-y}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x+3y times \frac{x-y}{x-y}.

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

Since \frac{\left(3x+3y\right)\left(x-y\right)}{x-y} and \frac{8xy}{x-y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+3y\right)\left(x-y\right)-8xy.

$$\frac{-3x^{2}+8xy+3y^{2}}{\frac{3x^{2}-3xy+3yx-3y^{2}-8xy}{x-y}}$$

Combine like terms in 3x^{2}-3xy+3yx-3y^{2}-8xy.

$$\frac{-3x^{2}+8xy+3y^{2}}{\frac{3x^{2}-3y^{2}-8xy}{x-y}}$$

Divide -3x^{2}+8xy+3y^{2} by \frac{3x^{2}-3y^{2}-8xy}{x-y} by multiplying -3x^{2}+8xy+3y^{2} by the reciprocal of \frac{3x^{2}-3y^{2}-8xy}{x-y}.

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

Extract the negative sign in -3x^{2}+8xy+3y^{2}.

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

Cancel out 3x^{2}-8xy-3y^{2} in both numerator and denominator.

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

To find the opposite of x-y, find the opposite of each term.

$$-x+y$$

Expand

$$y-x$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x-y times \frac{x-y}{x-y}.

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

Since \frac{\left(x-y\right)\left(x-y\right)}{x-y} and \frac{4y^{2}}{x-y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-y\right)\left(x-y\right)-4y^{2}.

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

Combine like terms in x^{2}-xy-xy+y^{2}-4y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x+y times \frac{x+y}{x+y}.

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

Since \frac{\left(x+y\right)\left(x+y\right)}{x+y} and \frac{4x^{2}}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+y\right)\left(x+y\right)-4x^{2}.

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

Combine like terms in x^{2}+xy+xy+y^{2}-4x^{2}.

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

Multiply \frac{x^{2}-3y^{2}-2xy}{x-y} times \frac{-3x^{2}+y^{2}+2xy}{x+y} by multiplying numerator times numerator and denominator times denominator.

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

Factor the expressions that are not already factored in \frac{\left(x^{2}-3y^{2}-2xy\right)\left(-3x^{2}+y^{2}+2xy\right)}{\left(x-y\right)\left(x+y\right)}.

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

Extract the negative sign in -x+y.

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

Cancel out \left(x+y\right)\left(x-y\right) in both numerator and denominator.

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

Expand the expression.

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

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

$$\frac{-3x^{2}+8xy+3y^{2}}{3x+3y-\frac{8xy}{x-y}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x+3y times \frac{x-y}{x-y}.

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

Since \frac{\left(3x+3y\right)\left(x-y\right)}{x-y} and \frac{8xy}{x-y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x+3y\right)\left(x-y\right)-8xy.

$$\frac{-3x^{2}+8xy+3y^{2}}{\frac{3x^{2}-3xy+3yx-3y^{2}-8xy}{x-y}}$$

Combine like terms in 3x^{2}-3xy+3yx-3y^{2}-8xy.

$$\frac{-3x^{2}+8xy+3y^{2}}{\frac{3x^{2}-3y^{2}-8xy}{x-y}}$$

Divide -3x^{2}+8xy+3y^{2} by \frac{3x^{2}-3y^{2}-8xy}{x-y} by multiplying -3x^{2}+8xy+3y^{2} by the reciprocal of \frac{3x^{2}-3y^{2}-8xy}{x-y}.

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

Extract the negative sign in -3x^{2}+8xy+3y^{2}.

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

Cancel out 3x^{2}-8xy-3y^{2} in both numerator and denominator.

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

To find the opposite of x-y, find the opposite of each term.

$$-x+y$$