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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-1. Factor x^{2}+2x+1.

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

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

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

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

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

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

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

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

$$\frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}}-\frac{4x}{\left(x-1\right)\left(x^{2}+2x+1\right)}$$

Factor \left(x-1\right)\left(x^{2}+2x+1\right).

$$\frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}}-\frac{4x}{\left(x-1\right)\left(x+1\right)^{2}}$$

Since \frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}} and \frac{4x}{\left(x-1\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+3x-4x}{\left(x-1\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{2}+3x-4x.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand \left(x+1\right)^{2}.

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

Expand

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

Short Solution Steps

Factor x^{2}-1. Factor x^{2}+2x+1.

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

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

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

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

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

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

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

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

$$\frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}}-\frac{4x}{\left(x-1\right)\left(x^{2}+2x+1\right)}$$

Factor \left(x-1\right)\left(x^{2}+2x+1\right).

$$\frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}}-\frac{4x}{\left(x-1\right)\left(x+1\right)^{2}}$$

Since \frac{x^{2}+3x}{\left(x-1\right)\left(x+1\right)^{2}} and \frac{4x}{\left(x-1\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+3x-4x}{\left(x-1\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{2}+3x-4x.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand \left(x+1\right)^{2}.

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