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

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

Evaluate

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

Short Solution Steps

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)\left(x+1\right)^{2}. Multiply \frac{1}{x-1} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x^{2}+1}{\left(x+1\right)^{2}} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{2}+2x+1-x^{3}+x^{2}-x+1.

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

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

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

Expand

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

Short Solution Steps

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)\left(x+1\right)^{2}. Multiply \frac{1}{x-1} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x^{2}+1}{\left(x+1\right)^{2}} times \frac{x-1}{x-1}.

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

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

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

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

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

Combine like terms in x^{2}+2x+1-x^{3}+x^{2}-x+1.

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

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

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