Solve -2/x^2-4+x-1/x^2-2x

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in -2x+\left(x-1\right)\left(x+2\right).

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

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

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

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

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

Cancel out x-2 in both numerator and denominator.

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

Expand x\left(x+2\right).

$$\frac{x+1}{x^{2}+2x}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

Do the multiplications in -2x+\left(x-1\right)\left(x+2\right).

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

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

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

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

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

Cancel out x-2 in both numerator and denominator.

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

Expand x\left(x+2\right).

$$\frac{x+1}{x^{2}+2x}$$