Solve 1/x-2-4x/x^2-1

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

Evaluate

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

Short Solution Steps

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

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

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

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

Factor x^{2}-1.

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

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

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

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

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

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

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

Combine like terms in x^{2}-1-2x^{3}+2x-4x^{2}.

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

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

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

Differentiate w.r.t. x

$$\frac{3x^{4}+6x^{2}-1}{\left(x\left(x^{2}-1\right)\right)^{2}}$$