Solve (-x/x+1+2x/1-x^2)*-3x^2+5x-2/x

$$( \frac { - x } { x + 1 } + \frac { 2 x } { 1 - x ^ { 2 } } ) \cdot \frac { - 3 x ^ { 2 } + 5 x - 2 } { x }$$

Evaluate

$$3x-2$$

Short Solution Steps

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{-x}{x-1}\times \frac{-3x^{2}+5x-2}{x}$$

Multiply \frac{-x}{x-1} times \frac{-3x^{2}+5x-2}{x} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

$$-\left(-1\right)\left(3x-2\right)$$

Expand the expression.

$$3x-2$$

Expand

$$3x-2$$

Short Solution Steps

Factor 1-x^{2}.

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -1-x.

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

Cancel out x+1 in both numerator and denominator.

$$\frac{-x}{x-1}\times \frac{-3x^{2}+5x-2}{x}$$

Multiply \frac{-x}{x-1} times \frac{-3x^{2}+5x-2}{x} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out x in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1-x.

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

Cancel out x-1 in both numerator and denominator.

$$-\left(-1\right)\left(3x-2\right)$$

Expand the expression.

$$3x-2$$