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

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

Evaluate

$$\frac{x^{4}+2x^{3}+9x^{2}+4x+2}{\left(x^{2}-1\right)\left(x^{2}+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+1 is \left(x-1\right)\left(x+1\right). Multiply \frac{3}{x-1} times \frac{x+1}{x+1}. Multiply \frac{x}{x+1} times \frac{x-1}{x-1}.

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

Since \frac{3\left(x+1\right)}{\left(x-1\right)\left(x+1\right)} and \frac{x\left(x-1\right)}{\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x+1\right)+x\left(x-1\right).

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

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

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

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 x^{2}+2 is \left(x-1\right)\left(x+1\right)\left(x^{2}+2\right). Multiply \frac{2x+3+x^{2}}{\left(x-1\right)\left(x+1\right)} times \frac{x^{2}+2}{x^{2}+2}. Multiply \frac{4}{x^{2}+2} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}.

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

Since \frac{\left(2x+3+x^{2}\right)\left(x^{2}+2\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+2\right)} and \frac{4\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x^{2}+2\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(2x+3+x^{2}\right)\left(x^{2}+2\right)+4\left(x-1\right)\left(x+1\right).

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

Combine like terms in 2x^{3}+4x+3x^{2}+6+x^{4}+2x^{2}+4x^{2}+4x-4x-4.

$$\frac{2x^{3}+4x+9x^{2}+2+x^{4}}{\left(x-1\right)\left(x+1\right)\left(x^{2}+2\right)}$$

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

$$\frac{2x^{3}+4x+9x^{2}+2+x^{4}}{x^{4}+x^{2}-2}$$

Differentiate w.r.t. x

$$-\frac{2\left(x^{6}+8x^{5}+5x^{4}+8x^{3}+8x^{2}+20x+4\right)}{\left(\left(x^{2}-1\right)\left(x^{2}+2\right)\right)^{2}}$$