Solve (x^2-y^2/x^2+2xy+y^2-x+y/x-y)div1/x+y

$$( \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 2 x y + y ^ { 2 } } - \frac { x + y } { x - y } ) \div \frac { 1 } { x + y }$$

Evaluate

$$-\frac{4xy}{x-y}$$

Short Solution Steps

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

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

Cancel out x+y in both numerator and denominator.

$$\frac{\frac{x-y}{x+y}-\frac{x+y}{x-y}}{\frac{1}{x+y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x+y}{x-y} times \frac{x+y}{x+y}.

$$\frac{\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

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

$$\frac{\frac{\left(x-y\right)\left(x-y\right)-\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

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

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

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

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

Divide \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by \frac{1}{x+y} by multiplying \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{1}{x+y}.

$$\frac{-4xy\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$$

Cancel out x+y in both numerator and denominator.

$$\frac{-4xy}{x-y}$$

Expand

$$-\frac{4xy}{x-y}$$

Short Solution Steps

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

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

Cancel out x+y in both numerator and denominator.

$$\frac{\frac{x-y}{x+y}-\frac{x+y}{x-y}}{\frac{1}{x+y}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+y and x-y is \left(x+y\right)\left(x-y\right). Multiply \frac{x-y}{x+y} times \frac{x-y}{x-y}. Multiply \frac{x+y}{x-y} times \frac{x+y}{x+y}.

$$\frac{\frac{\left(x-y\right)\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}-\frac{\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

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

$$\frac{\frac{\left(x-y\right)\left(x-y\right)-\left(x+y\right)\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

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

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

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

$$\frac{\frac{-4xy}{\left(x+y\right)\left(x-y\right)}}{\frac{1}{x+y}}$$

Divide \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by \frac{1}{x+y} by multiplying \frac{-4xy}{\left(x+y\right)\left(x-y\right)} by the reciprocal of \frac{1}{x+y}.

$$\frac{-4xy\left(x+y\right)}{\left(x+y\right)\left(x-y\right)}$$

Cancel out x+y in both numerator and denominator.

$$\frac{-4xy}{x-y}$$