Solve x/x^2-4+2/x^2-2x-1/x^2+2x):(1-4/x+2+frac{3}{x})

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

Evaluate

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

Short Solution Steps

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

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

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) and x\left(x-2\right) is x\left(x-2\right)\left(x+2\right). Multiply \frac{x}{\left(x-2\right)\left(x+2\right)} times \frac{x}{x}. Multiply \frac{2}{x\left(x-2\right)} times \frac{x+2}{x+2}.

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

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

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

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

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

Factor x^{2}+2x.

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

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

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

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

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

Do the multiplications in x^{2}+2x+4-\left(x-2\right).

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

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

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

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

$$\frac{x^{2}+x+6}{x^{3}-4x}$$

Differentiate w.r.t. x

$$\frac{24-22x^{2}-2x^{3}-x^{4}}{\left(x\left(x^{2}-4\right)\right)^{2}}$$