Solve 1/x-2+1-2x^2/x^3-8+x/x^2+2x+4

$$\frac { 1 } { x - 2 } + \frac { 1 - 2 x ^ { 2 } } { x ^ { 3 } - 8 } + \frac { x } { x ^ { 2 } + 2 x + 4 }$$

Evaluate

$$\frac{5}{x^{3}-8}$$

Short Solution Steps

Factor x^{3}-8.

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

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

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

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

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

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

$$\frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{x}{x^{2}+2x+4}$$

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

$$\frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Since \frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+2x+5+x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Do the multiplications in -x^{2}+2x+5+x\left(x-2\right).

$$\frac{-x^{2}+2x+5+x^{2}-2x}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Combine like terms in -x^{2}+2x+5+x^{2}-2x.

$$\frac{5}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Expand \left(x-2\right)\left(x^{2}+2x+4\right).

$$\frac{5}{x^{3}-8}$$

Expand

$$\frac{5}{x^{3}-8}$$

Short Solution Steps

Factor x^{3}-8.

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

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

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

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

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

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

$$\frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{x}{x^{2}+2x+4}$$

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

$$\frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Since \frac{-x^{2}+2x+5}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+2x+5+x\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Do the multiplications in -x^{2}+2x+5+x\left(x-2\right).

$$\frac{-x^{2}+2x+5+x^{2}-2x}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Combine like terms in -x^{2}+2x+5+x^{2}-2x.

$$\frac{5}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Expand \left(x-2\right)\left(x^{2}+2x+4\right).

$$\frac{5}{x^{3}-8}$$