Solve (2x/x^2+2x+1-x/x-1):(x/x^2-1-2x)]:(9/4x^2-6+frac{1}{2})

$$( \frac { 2 x } { x ^ { 2 } + 2 x + 1 } - \frac { x } { x - 1 } ) : ( \frac { x } { x ^ { 2 } - 1 } - 2 x ) ] : ( \frac { 9 } { 4 x ^ { 2 } - 6 } + \frac { 1 } { 2 } )$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Cancel out \left(x-1\right)\left(x+1\right) in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x in both numerator and denominator.

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

Expand the expression.

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