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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Since \frac{x^{4}-2x^{3}-x^{2}+1}{x^{2}} and \frac{2x}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Factor

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