Solve 2/x-2-12x/x^3-8-x-2/x^2+2x+4

$$\frac { 2 } { x - 2 } - \frac { 12 x } { x ^ { 3 } - 8 } - \frac { x - 2 } { x ^ { 2 } + 2 x + 4 }$$

Evaluate

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

Short Solution Steps

Factor x^{3}-8.

$$\frac{2}{x-2}-\frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

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{2}{x-2} times \frac{x^{2}+2x+4}{x^{2}+2x+4}.

$$\frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Since \frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(x^{2}+2x+4\right)-12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Do the multiplications in 2\left(x^{2}+2x+4\right)-12x.

$$\frac{2x^{2}+4x+8-12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Combine like terms in 2x^{2}+4x+8-12x.

$$\frac{2x^{2}-8x+8}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Factor the expressions that are not already factored in \frac{2x^{2}-8x+8}{\left(x-2\right)\left(x^{2}+2x+4\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

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

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

Combine like terms in 2x-4-x+2.

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

Expand

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

Short Solution Steps

Factor x^{3}-8.

$$\frac{2}{x-2}-\frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

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{2}{x-2} times \frac{x^{2}+2x+4}{x^{2}+2x+4}.

$$\frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Since \frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{12x}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(x^{2}+2x+4\right)-12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Do the multiplications in 2\left(x^{2}+2x+4\right)-12x.

$$\frac{2x^{2}+4x+8-12x}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Combine like terms in 2x^{2}+4x+8-12x.

$$\frac{2x^{2}-8x+8}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{x-2}{x^{2}+2x+4}$$

Factor the expressions that are not already factored in \frac{2x^{2}-8x+8}{\left(x-2\right)\left(x^{2}+2x+4\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

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

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

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

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

Combine like terms in 2x-4-x+2.

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