Solve 2/x^4-4x^2-1/x^3-x^2-4x+4

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

Evaluate

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

Short Solution Steps

Factor x^{4}-4x^{2}. Factor x^{3}-x^{2}-4x+4.

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

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

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

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

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

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

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

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

$$\frac{2x-2-x^{2}}{x^{5}-x^{4}-4x^{3}+4x^{2}}$$

Differentiate w.r.t. x

$$\frac{3x^{5}-10x^{4}+12x^{3}+8x^{2}-32x+16}{x^{3}\left(\left(x-1\right)\left(x^{2}-4\right)\right)^{2}}$$