Solve x-1/x-x-2/x+1+2x^2-x/x^2+2x+1

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

Evaluate

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

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{x-2}{x+1} times \frac{x}{x}.

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

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

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

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

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

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

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

Factor x^{2}+2x+1.

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

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

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

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

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

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

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

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

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

Expand x\left(x+1\right)^{2}.

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

Expand

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

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{x-2}{x+1} times \frac{x}{x}.

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

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

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

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

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

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

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

Factor x^{2}+2x+1.

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

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

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

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

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

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

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

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

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

Expand x\left(x+1\right)^{2}.

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