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

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

Evaluate

$$-\frac{x\left(x^{3}+3x^{2}+3x+10\right)}{\left(x+1\right)^{2}}$$

Short Solution Steps

Express x\times \frac{2-x}{x+1} as a single fraction.

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

To raise \frac{2-x}{x+1} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}-4 times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x+1\right)^{2} is \left(x+1\right)^{2}. Multiply \frac{-x^{3}-2x^{2}-2x-4}{x+1} times \frac{x+1}{x+1}.

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

Since \frac{\left(-x^{3}-2x^{2}-2x-4\right)\left(x+1\right)}{\left(x+1\right)^{2}} and \frac{\left(2-x\right)^{2}}{\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(-x^{3}-2x^{2}-2x-4\right)\left(x+1\right)+\left(2-x\right)^{2}.

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

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

$$\frac{-x^{4}-3x^{3}-3x^{2}-10x}{\left(x+1\right)^{2}}$$

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

$$\frac{-x^{4}-3x^{3}-3x^{2}-10x}{x^{2}+2x+1}$$

Expand

$$-\frac{x^{4}+3x^{3}+3x^{2}+10x}{\left(x+1\right)^{2}}$$

Short Solution Steps

Express x\times \frac{2-x}{x+1} as a single fraction.

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

To raise \frac{2-x}{x+1} to a power, raise both numerator and denominator to the power and then divide.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}-4 times \frac{x+1}{x+1}.

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and \left(x+1\right)^{2} is \left(x+1\right)^{2}. Multiply \frac{-x^{3}-2x^{2}-2x-4}{x+1} times \frac{x+1}{x+1}.

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

Since \frac{\left(-x^{3}-2x^{2}-2x-4\right)\left(x+1\right)}{\left(x+1\right)^{2}} and \frac{\left(2-x\right)^{2}}{\left(x+1\right)^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(-x^{3}-2x^{2}-2x-4\right)\left(x+1\right)+\left(2-x\right)^{2}.

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

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

$$\frac{-x^{4}-3x^{3}-3x^{2}-10x}{\left(x+1\right)^{2}}$$

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

$$\frac{-x^{4}-3x^{3}-3x^{2}-10x}{x^{2}+2x+1}$$