Solve =2x^2+4x/x^3-4x+x^2-4/x^2+2x+2/2-x

$$= \frac { 2 x ^ { 2 } + 4 x } { x ^ { 3 } - 4 x } + \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 2 x } + \frac { 2 } { 2 - x }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x^{2}+4x}{x^{3}-4x}.

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

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

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

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

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

Cancel out x+2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Since \frac{-2x+x^{2}+4}{x\left(x-2\right)} and \frac{2\left(-1\right)x}{x\left(x-2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -2x+x^{2}+4+2\left(-1\right)x.

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

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

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

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

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

Cancel out x-2 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{2x^{2}+4x}{x^{3}-4x}.

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

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

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

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

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

Cancel out x+2 in both numerator and denominator.

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

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

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

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

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

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

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

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

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

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

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

Since \frac{-2x+x^{2}+4}{x\left(x-2\right)} and \frac{2\left(-1\right)x}{x\left(x-2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -2x+x^{2}+4+2\left(-1\right)x.

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

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

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

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

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

Cancel out x-2 in both numerator and denominator.

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