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

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

Evaluate

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

Short Solution Steps

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

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

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

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

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