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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{2}+x+2 by each term of x-1.

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

Combine -x^{2} and x^{2} to get 0.

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

Combine -x and 2x to get x.

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

Apply the distributive property by multiplying each term of x+1 by each term of x^{2}-x-1.

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

Combine -x^{2} and x^{2} to get 0.

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

Combine -x and -x to get -2x.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Apply the distributive property by multiplying each term of x^{2}+x+2 by each term of x-1.

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

Combine -x^{2} and x^{2} to get 0.

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

Combine -x and 2x to get x.

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

Apply the distributive property by multiplying each term of x+1 by each term of x^{2}-x-1.

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

Combine -x^{2} and x^{2} to get 0.

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

Combine -x and -x to get -2x.

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