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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+x+x+1+x.

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

Factor x^{2}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x\left(x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{x^{2}+3x+1}{x\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{2-x^{2}}{x\left(x-1\right)} times \frac{x+1}{x+1}.

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

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

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

Do the multiplications in \left(x^{2}+3x+1\right)\left(x-1\right)+\left(2-x^{2}\right)\left(x+1\right).

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

Combine like terms in x^{3}-x^{2}+3x^{2}-3x+x-1+2x+2-x^{3}-x^{2}.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{4}+2x^{3}+x^{2}}{x^{3}-x^{2}+x-1}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+x+x+1+x.

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

Factor x^{2}-x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x\left(x-1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{x^{2}+3x+1}{x\left(x+1\right)} times \frac{x-1}{x-1}. Multiply \frac{2-x^{2}}{x\left(x-1\right)} times \frac{x+1}{x+1}.

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

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

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

Do the multiplications in \left(x^{2}+3x+1\right)\left(x-1\right)+\left(2-x^{2}\right)\left(x+1\right).

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

Combine like terms in x^{3}-x^{2}+3x^{2}-3x+x-1+2x+2-x^{3}-x^{2}.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x-1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{4}+2x^{3}+x^{2}}{x^{3}-x^{2}+x-1}$$