Solve 2/x-2+2x+3/x^2+2x+4-6x+12/x^3-8

$$\frac { 2 } { x - 2 } + \frac { 2 x + 3 } { x ^ { 2 } + 2 x + 4 } - \frac { 6 x + 12 } { x ^ { 3 } - 8 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x^{2}+2x+4 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}. Multiply \frac{2x+3}{x^{2}+2x+4} times \frac{x-2}{x-2}.

$$\frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{\left(2x+3\right)\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

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

$$\frac{2\left(x^{2}+2x+4\right)+\left(2x+3\right)\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

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

$$\frac{2x^{2}+4x+8+2x^{2}-4x+3x-6}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

Combine like terms in 2x^{2}+4x+8+2x^{2}-4x+3x-6.

$$\frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

Factor x^{3}-8.

$$\frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Since \frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{6x+12}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x^{2}+3x+2-\left(6x+12\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Do the multiplications in 4x^{2}+3x+2-\left(6x+12\right).

$$\frac{4x^{2}+3x+2-6x-12}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Combine like terms in 4x^{2}+3x+2-6x-12.

$$\frac{4x^{2}-3x-10}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-3x-10}{\left(x-2\right)\left(x^{2}+2x+4\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x^{2}+2x+4 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}. Multiply \frac{2x+3}{x^{2}+2x+4} times \frac{x-2}{x-2}.

$$\frac{2\left(x^{2}+2x+4\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}+\frac{\left(2x+3\right)\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

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

$$\frac{2\left(x^{2}+2x+4\right)+\left(2x+3\right)\left(x-2\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

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

$$\frac{2x^{2}+4x+8+2x^{2}-4x+3x-6}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

Combine like terms in 2x^{2}+4x+8+2x^{2}-4x+3x-6.

$$\frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{x^{3}-8}$$

Factor x^{3}-8.

$$\frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)}-\frac{6x+12}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Since \frac{4x^{2}+3x+2}{\left(x-2\right)\left(x^{2}+2x+4\right)} and \frac{6x+12}{\left(x-2\right)\left(x^{2}+2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x^{2}+3x+2-\left(6x+12\right)}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Do the multiplications in 4x^{2}+3x+2-\left(6x+12\right).

$$\frac{4x^{2}+3x+2-6x-12}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Combine like terms in 4x^{2}+3x+2-6x-12.

$$\frac{4x^{2}-3x-10}{\left(x-2\right)\left(x^{2}+2x+4\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-3x-10}{\left(x-2\right)\left(x^{2}+2x+4\right)}.

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

Cancel out x-2 in both numerator and denominator.

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