Solve 2/x+1-x+2/x^2+x-6x/x^2-1=

$$\frac{ 2 }{ x+1 } - \frac{ x+2 }{ { x }^{ 2 } +x } - \frac{ 6x }{ { x }^{ 2 } -1 } =$$

Evaluate

$$-\frac{5x-2}{x\left(x-1\right)}$$

Short Solution Steps

Factor x^{2}+x.

$$\frac{2}{x+1}-\frac{x+2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x}{x\left(x+1\right)}-\frac{x+2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x-\left(x+2\right)}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x-x-2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

Combine like terms in 2x-x-2.

$$\frac{x-2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

Factor x^{2}-1.

$$\frac{x-2}{x\left(x+1\right)}-\frac{6x}{\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and \left(x-1\right)\left(x+1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{x-2}{x\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{6x}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}.

$$\frac{\left(x-2\right)\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)}-\frac{6xx}{x\left(x-1\right)\left(x+1\right)}$$

Since \frac{\left(x-2\right)\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{6xx}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(x-1\right)-6xx}{x\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in \left(x-2\right)\left(x-1\right)-6xx.

$$\frac{x^{2}-x-2x+2-6x^{2}}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in x^{2}-x-2x+2-6x^{2}.

$$\frac{-5x^{2}-3x+2}{x\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{-5x^{2}-3x+2}{x\left(x-1\right)\left(x+1\right)}.

$$\frac{\left(-x-1\right)\left(5x-2\right)}{x\left(x-1\right)\left(x+1\right)}$$

Extract the negative sign in -1-x.

$$\frac{-\left(5x-2\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{-\left(5x-2\right)}{x\left(x-1\right)}$$

Expand x\left(x-1\right).

$$\frac{-\left(5x-2\right)}{x^{2}-x}$$

To find the opposite of 5x-2, find the opposite of each term.

$$\frac{-5x+2}{x^{2}-x}$$

Expand

$$\frac{2-5x}{x\left(x-1\right)}$$

Short Solution Steps

Factor x^{2}+x.

$$\frac{2}{x+1}-\frac{x+2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x}{x\left(x+1\right)}-\frac{x+2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x-\left(x+2\right)}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

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

$$\frac{2x-x-2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

Combine like terms in 2x-x-2.

$$\frac{x-2}{x\left(x+1\right)}-\frac{6x}{x^{2}-1}$$

Factor x^{2}-1.

$$\frac{x-2}{x\left(x+1\right)}-\frac{6x}{\left(x-1\right)\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and \left(x-1\right)\left(x+1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{x-2}{x\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{6x}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}.

$$\frac{\left(x-2\right)\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)}-\frac{6xx}{x\left(x-1\right)\left(x+1\right)}$$

Since \frac{\left(x-2\right)\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{6xx}{x\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(x-1\right)-6xx}{x\left(x-1\right)\left(x+1\right)}$$

Do the multiplications in \left(x-2\right)\left(x-1\right)-6xx.

$$\frac{x^{2}-x-2x+2-6x^{2}}{x\left(x-1\right)\left(x+1\right)}$$

Combine like terms in x^{2}-x-2x+2-6x^{2}.

$$\frac{-5x^{2}-3x+2}{x\left(x-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{-5x^{2}-3x+2}{x\left(x-1\right)\left(x+1\right)}.

$$\frac{\left(-x-1\right)\left(5x-2\right)}{x\left(x-1\right)\left(x+1\right)}$$

Extract the negative sign in -1-x.

$$\frac{-\left(5x-2\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{-\left(5x-2\right)}{x\left(x-1\right)}$$

Expand x\left(x-1\right).

$$\frac{-\left(5x-2\right)}{x^{2}-x}$$

To find the opposite of 5x-2, find the opposite of each term.

$$\frac{-5x+2}{x^{2}-x}$$