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

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

Evaluate

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

Short Solution Steps

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

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

Cancel out x+2y in both numerator and denominator.

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

Factor 4y^{2}-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2y and \left(x+2y\right)\left(-x+2y\right) is \left(x+2y\right)\left(-x+2y\right). Multiply \frac{1}{x+2y} times \frac{-x+2y}{-x+2y}.

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

Since \frac{-x+2y}{\left(x+2y\right)\left(-x+2y\right)} and \frac{2x}{\left(x+2y\right)\left(-x+2y\right)} have the same denominator, add them by adding their numerators.

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

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

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

Cancel out x+2y in both numerator and denominator.

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

Expand

$$\frac{2}{4y-2x}$$

Short Solution Steps

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

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

Cancel out x+2y in both numerator and denominator.

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

Factor 4y^{2}-x^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2y and \left(x+2y\right)\left(-x+2y\right) is \left(x+2y\right)\left(-x+2y\right). Multiply \frac{1}{x+2y} times \frac{-x+2y}{-x+2y}.

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

Since \frac{-x+2y}{\left(x+2y\right)\left(-x+2y\right)} and \frac{2x}{\left(x+2y\right)\left(-x+2y\right)} have the same denominator, add them by adding their numerators.

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

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

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

Cancel out x+2y in both numerator and denominator.

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