Solve left(x+1/x-1+1right)divx/x^2-2x+1+2-2x/x^2-1

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

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

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

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

Combine like terms in x+1+x-1.

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

Divide \frac{2x}{x-1} by \frac{x}{x^{2}-2x+1} by multiplying \frac{2x}{x-1} by the reciprocal of \frac{x}{x^{2}-2x+1}.

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

Cancel out x in both numerator and denominator.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand the expression.

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

Factor the expressions that are not already factored in \frac{2-2x}{x^{2}-1}.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-2 times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

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

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

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

Combine like terms in x+1+x-1.

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

Divide \frac{2x}{x-1} by \frac{x}{x^{2}-2x+1} by multiplying \frac{2x}{x-1} by the reciprocal of \frac{x}{x^{2}-2x+1}.

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

Cancel out x in both numerator and denominator.

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

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

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

Cancel out x-1 in both numerator and denominator.

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

Expand the expression.

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

Factor the expressions that are not already factored in \frac{2-2x}{x^{2}-1}.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x-2 times \frac{x+1}{x+1}.

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

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

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

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

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

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

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