Solve x+2/(1-x)^2+2/x+x^2-1/1-x=

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

Evaluate

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

Short Solution Steps

Factor x+x^{2}.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(-x+1\right)^{2} and 1-x is x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}. Multiply \frac{x^{3}+5x^{2}-2x+2}{x\left(x+1\right)\left(-x+1\right)^{2}} times \frac{-x+1}{-x+1}. Multiply \frac{1}{1-x} times \frac{x\left(x+1\right)\left(-x+1\right)^{2}}{x\left(x+1\right)\left(-x+1\right)^{2}}.

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

Since \frac{\left(x^{3}+5x^{2}-2x+2\right)\left(-x+1\right)}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}} and \frac{x\left(x+1\right)\left(-x+1\right)^{2}}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{3}+5x^{2}-2x+2\right)\left(-x+1\right)-x\left(x+1\right)\left(-x+1\right)^{2}.

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

Combine like terms in -x^{4}+x^{3}-5x^{3}+5x^{2}+2x^{2}-2x-2x+2-x^{4}+2x^{3}-x^{2}-x^{3}+2x^{2}-x.

$$\frac{-2x^{4}-3x^{3}+8x^{2}-5x+2}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-2x^{4}-3x^{3}+8x^{2}-5x+2}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}}.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

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

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

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

Use the distributive property to multiply -2 by -x^{3}-\frac{5}{2}x^{2}+\frac{3}{2}x-1.

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

Expand

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

Short Solution Steps

Factor x+x^{2}.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right)\left(-x+1\right)^{2} and 1-x is x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}. Multiply \frac{x^{3}+5x^{2}-2x+2}{x\left(x+1\right)\left(-x+1\right)^{2}} times \frac{-x+1}{-x+1}. Multiply \frac{1}{1-x} times \frac{x\left(x+1\right)\left(-x+1\right)^{2}}{x\left(x+1\right)\left(-x+1\right)^{2}}.

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

Since \frac{\left(x^{3}+5x^{2}-2x+2\right)\left(-x+1\right)}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}} and \frac{x\left(x+1\right)\left(-x+1\right)^{2}}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x^{3}+5x^{2}-2x+2\right)\left(-x+1\right)-x\left(x+1\right)\left(-x+1\right)^{2}.

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

Combine like terms in -x^{4}+x^{3}-5x^{3}+5x^{2}+2x^{2}-2x-2x+2-x^{4}+2x^{3}-x^{2}-x^{3}+2x^{2}-x.

$$\frac{-2x^{4}-3x^{3}+8x^{2}-5x+2}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-2x^{4}-3x^{3}+8x^{2}-5x+2}{x\left(x+1\right)\left(-x+1\right)\left(-x+1\right)^{2}}.

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

Extract the negative sign in -1+x.

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

Cancel out -x+1 in both numerator and denominator.

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

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

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

Use the distributive property to multiply -2 by -x^{3}-\frac{5}{2}x^{2}+\frac{3}{2}x-1.

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