Solve (2-1/x+1+1/x-1)*x^3-5x^2/x^2-4x-5div(x-x/1-x^2)

$$( 2 - \frac { 1 } { x + 1 } + \frac { 1 } { x - 1 } ) \cdot \frac { x ^ { 3 } - 5 x ^ { 2 } } { x ^ { 2 } - 4 x - 5 } \div ( x - \frac { x } { 1 - x ^ { 2 } } )$$

Evaluate

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

Short Solution Steps

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

$$\frac{\left(\frac{2\left(x+1\right)}{x+1}-\frac{1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{2\left(x+1\right)-1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{2x+2-1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

Combine like terms in 2x+2-1.

$$\frac{\left(\frac{2x+1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{\left(2x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{\left(2x+1\right)\left(x-1\right)+x+1}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{2x^{2}-2x-1+x+x+1}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{2x^{2}}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

Factor the expressions that are not already factored in \frac{x^{3}-5x^{2}}{x^{2}-4x-5}.

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

Cancel out x-5 in both numerator and denominator.

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

Multiply \frac{2x^{2}}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

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

Divide \frac{2x^{2}x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+1\right)} by \frac{-x^{3}}{\left(x-1\right)\left(-x-1\right)} by multiplying \frac{2x^{2}x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+1\right)} by the reciprocal of \frac{-x^{3}}{\left(x-1\right)\left(-x-1\right)}.

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

Extract the negative sign in -x-1.

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

Cancel out -x\left(x-1\right)\left(x+1\right)x^{2} in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

$$\frac{\left(\frac{2\left(x+1\right)}{x+1}-\frac{1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{2\left(x+1\right)-1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{2x+2-1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

Combine like terms in 2x+2-1.

$$\frac{\left(\frac{2x+1}{x+1}+\frac{1}{x-1}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\left(\frac{\left(2x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)}+\frac{x+1}{\left(x-1\right)\left(x+1\right)}\right)\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{\left(2x+1\right)\left(x-1\right)+x+1}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{2x^{2}-2x-1+x+x+1}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

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

$$\frac{\frac{2x^{2}}{\left(x-1\right)\left(x+1\right)}\times \frac{x^{3}-5x^{2}}{x^{2}-4x-5}}{x-\frac{x}{1-x^{2}}}$$

Factor the expressions that are not already factored in \frac{x^{3}-5x^{2}}{x^{2}-4x-5}.

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

Cancel out x-5 in both numerator and denominator.

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

Multiply \frac{2x^{2}}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

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

Divide \frac{2x^{2}x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+1\right)} by \frac{-x^{3}}{\left(x-1\right)\left(-x-1\right)} by multiplying \frac{2x^{2}x^{2}}{\left(x-1\right)\left(x+1\right)\left(x+1\right)} by the reciprocal of \frac{-x^{3}}{\left(x-1\right)\left(-x-1\right)}.

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

Extract the negative sign in -x-1.

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

Cancel out -x\left(x-1\right)\left(x+1\right)x^{2} in both numerator and denominator.

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