Solve x+1/x-1-x/x+1+4x/1+x^2

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

Evaluate

$$\frac{2x^{4}+5x^{3}+7x^{2}+x+1}{x\left(x+1\right)\left(x^{2}+1\right)}$$

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and 1+x^{2} is x\left(x+1\right)\left(x^{2}+1\right). Multiply \frac{x+2x^{2}+1}{x\left(x+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{4x}{1+x^{2}} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

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

Since \frac{\left(x+2x^{2}+1\right)\left(x^{2}+1\right)}{x\left(x+1\right)\left(x^{2}+1\right)} and \frac{4xx\left(x+1\right)}{x\left(x+1\right)\left(x^{2}+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+2x^{2}+1\right)\left(x^{2}+1\right)+4xx\left(x+1\right).

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

Combine like terms in x^{3}+x+2x^{4}+2x^{2}+x^{2}+1+4x^{3}+4x^{2}.

$$\frac{5x^{3}+x+2x^{4}+7x^{2}+1}{x\left(x+1\right)\left(x^{2}+1\right)}$$

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

$$\frac{5x^{3}+x+2x^{4}+7x^{2}+1}{x^{4}+x^{3}+x^{2}+x}$$

Expand

$$\frac{2x^{4}+5x^{3}+7x^{2}+x+1}{x\left(x+1\right)\left(x^{2}+1\right)}$$

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and 1+x^{2} is x\left(x+1\right)\left(x^{2}+1\right). Multiply \frac{x+2x^{2}+1}{x\left(x+1\right)} times \frac{x^{2}+1}{x^{2}+1}. Multiply \frac{4x}{1+x^{2}} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

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

Since \frac{\left(x+2x^{2}+1\right)\left(x^{2}+1\right)}{x\left(x+1\right)\left(x^{2}+1\right)} and \frac{4xx\left(x+1\right)}{x\left(x+1\right)\left(x^{2}+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x+2x^{2}+1\right)\left(x^{2}+1\right)+4xx\left(x+1\right).

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

Combine like terms in x^{3}+x+2x^{4}+2x^{2}+x^{2}+1+4x^{3}+4x^{2}.

$$\frac{5x^{3}+x+2x^{4}+7x^{2}+1}{x\left(x+1\right)\left(x^{2}+1\right)}$$

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

$$\frac{5x^{3}+x+2x^{4}+7x^{2}+1}{x^{4}+x^{3}+x^{2}+x}$$