Solve 1-x-y/x+2ydivx^2-y^2/x^2+4xy+4y^2

$$1 - \frac { x - y } { x + 2 y } \div \frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 2 } + 4 x y + 4 y ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Divide \frac{x-y}{x+2y} by \frac{x^{2}-y^{2}}{x^{2}+4xy+4y^{2}} by multiplying \frac{x-y}{x+2y} by the reciprocal of \frac{x^{2}-y^{2}}{x^{2}+4xy+4y^{2}}.

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

Factor the expressions that are not already factored in \frac{\left(x-y\right)\left(x^{2}+4xy+4y^{2}\right)}{\left(x+2y\right)\left(x^{2}-y^{2}\right)}.

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

Cancel out \left(x-y\right)\left(x+2y\right) in both numerator and denominator.

$$1-\frac{x+2y}{x+y}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{x+y}{x+y}-\frac{x+2y}{x+y}$$

Since \frac{x+y}{x+y} and \frac{x+2y}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+y-\left(x+2y\right).

$$\frac{x+y-x-2y}{x+y}$$

Combine like terms in x+y-x-2y.

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

Expand

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

Short Solution Steps

Divide \frac{x-y}{x+2y} by \frac{x^{2}-y^{2}}{x^{2}+4xy+4y^{2}} by multiplying \frac{x-y}{x+2y} by the reciprocal of \frac{x^{2}-y^{2}}{x^{2}+4xy+4y^{2}}.

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

Factor the expressions that are not already factored in \frac{\left(x-y\right)\left(x^{2}+4xy+4y^{2}\right)}{\left(x+2y\right)\left(x^{2}-y^{2}\right)}.

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

Cancel out \left(x-y\right)\left(x+2y\right) in both numerator and denominator.

$$1-\frac{x+2y}{x+y}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+y}{x+y}.

$$\frac{x+y}{x+y}-\frac{x+2y}{x+y}$$

Since \frac{x+y}{x+y} and \frac{x+2y}{x+y} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x+y-\left(x+2y\right).

$$\frac{x+y-x-2y}{x+y}$$

Combine like terms in x+y-x-2y.

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