Solve x-1/x+frac{2{x}-x^2+2}{x-x-2/x+1}

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Combine x and -x to get 0.

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

Multiply \frac{\left(x-1\right)x}{x^{2}-x^{3}+2x+2} times \frac{-2}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply x-1 by x.

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

Use the distributive property to multiply x^{2}-x by -2.

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

Use the distributive property to multiply x^{2}-x^{3}+2x+2 by x+1 and combine like terms.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

Combine x and -x to get 0.

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

Multiply \frac{\left(x-1\right)x}{x^{2}-x^{3}+2x+2} times \frac{-2}{x+1} by multiplying numerator times numerator and denominator times denominator.

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

Use the distributive property to multiply x-1 by x.

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

Use the distributive property to multiply x^{2}-x by -2.

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

Use the distributive property to multiply x^{2}-x^{3}+2x+2 by x+1 and combine like terms.

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