Solve =(x+2/x+1+2x/2-x+x^2+4x+6/x^2-x-2):x^2+1/2x^2-4x

$$= ( \frac { x + 2 } { x + 1 } + \frac { 2 x } { 2 - x } + \frac { x ^ { 2 } + 4 x + 6 } { x ^ { 2 } - x - 2 } ) : \frac { x ^ { 2 } + 1 } { 2 x ^ { 2 } - 4 x }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\frac{\left(x+2\right)\left(-x+2\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{2x\left(x+1\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{\left(x+2\right)\left(-x+2\right)+2x\left(x+1\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{-x^{2}+2x-2x+4+2x^{2}+2x}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{x^{2}+2x+4}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Factor x^{2}-x-2.

$$\frac{\frac{x^{2}+2x+4}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{-\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x+1\right)}+\frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Since \frac{-\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x+1\right)} and \frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-\left(x^{2}+2x+4\right)+x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Do the multiplications in -\left(x^{2}+2x+4\right)+x^{2}+4x+6.

$$\frac{\frac{-x^{2}-2x-4+x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Combine like terms in -x^{2}-2x-4+x^{2}+4x+6.

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

Factor the expressions that are not already factored in \frac{2x+2}{\left(x-2\right)\left(x+1\right)}.

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

Cancel out x+1 in both numerator and denominator.

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

Divide \frac{2}{x-2} by \frac{x^{2}+1}{2x^{2}-4x} by multiplying \frac{2}{x-2} by the reciprocal of \frac{x^{2}+1}{2x^{2}-4x}.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

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

$$\frac{\frac{\left(x+2\right)\left(-x+2\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{2x\left(x+1\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{\left(x+2\right)\left(-x+2\right)+2x\left(x+1\right)}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{-x^{2}+2x-2x+4+2x^{2}+2x}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{x^{2}+2x+4}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{x^{2}-x-2}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Factor x^{2}-x-2.

$$\frac{\frac{x^{2}+2x+4}{\left(x+1\right)\left(-x+2\right)}+\frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

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

$$\frac{\frac{-\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x+1\right)}+\frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Since \frac{-\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x+1\right)} and \frac{x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-\left(x^{2}+2x+4\right)+x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Do the multiplications in -\left(x^{2}+2x+4\right)+x^{2}+4x+6.

$$\frac{\frac{-x^{2}-2x-4+x^{2}+4x+6}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+1}{2x^{2}-4x}}$$

Combine like terms in -x^{2}-2x-4+x^{2}+4x+6.

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

Factor the expressions that are not already factored in \frac{2x+2}{\left(x-2\right)\left(x+1\right)}.

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

Cancel out x+1 in both numerator and denominator.

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

Divide \frac{2}{x-2} by \frac{x^{2}+1}{2x^{2}-4x} by multiplying \frac{2}{x-2} by the reciprocal of \frac{x^{2}+1}{2x^{2}-4x}.

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

Factor the expressions that are not already factored.

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

Cancel out x-2 in both numerator and denominator.

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

Expand the expression.

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