Solve x+1/x^2-9-2x/x^2+2x-15+x^2+3x+9/x^3-27

$$\frac { x + 1 } { x ^ { 2 } - 9 } - \frac { 2 x } { x ^ { 2 } + 2 x - 15 } + \frac { x ^ { 2 } + 3 x + 9 } { x ^ { 3 } - 27 }$$

Evaluate

$$\frac{4\left(2x+5\right)}{\left(x+5\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+3x+9}{x^{3}-27}.

$$\frac{x+1}{x^{2}-9}-\frac{2x}{x^{2}+2x-15}+\frac{x^{2}+3x+9}{\left(x-3\right)\left(x^{2}+3x+9\right)}$$

Cancel out x^{2}+3x+9 in both numerator and denominator.

$$\frac{x+1}{x^{2}-9}-\frac{2x}{x^{2}+2x-15}+\frac{1}{x-3}$$

Factor x^{2}-9. Factor x^{2}+2x-15.

$$\frac{x+1}{\left(x-3\right)\left(x+3\right)}-\frac{2x}{\left(x-3\right)\left(x+5\right)}+\frac{1}{x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+3\right)\left(x+5\right). Multiply \frac{x+1}{\left(x-3\right)\left(x+3\right)} times \frac{x+5}{x+5}. Multiply \frac{2x}{\left(x-3\right)\left(x+5\right)} times \frac{x+3}{x+3}.

$$\frac{\left(x+1\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}-\frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{1}{x-3}$$

Since \frac{\left(x+1\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} and \frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x+5\right)-2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{1}{x-3}$$

Do the multiplications in \left(x+1\right)\left(x+5\right)-2x\left(x+3\right).

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

Combine like terms in x^{2}+5x+x+5-2x^{2}-6x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right)\left(x+5\right) and x-3 is \left(x-3\right)\left(x+3\right)\left(x+5\right). Multiply \frac{1}{x-3} times \frac{\left(x+3\right)\left(x+5\right)}{\left(x+3\right)\left(x+5\right)}.

$$\frac{-x^{2}+5}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Since \frac{-x^{2}+5}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} and \frac{\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+5+\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Do the multiplications in -x^{2}+5+\left(x+3\right)\left(x+5\right).

$$\frac{-x^{2}+5+x^{2}+5x+3x+15}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Combine like terms in -x^{2}+5+x^{2}+5x+3x+15.

$$\frac{20+8x}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Expand \left(x-3\right)\left(x+3\right)\left(x+5\right).

$$\frac{20+8x}{x^{3}+5x^{2}-9x-45}$$

Expand

$$\frac{4\left(2x+5\right)}{\left(x+5\right)\left(x^{2}-9\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}+3x+9}{x^{3}-27}.

$$\frac{x+1}{x^{2}-9}-\frac{2x}{x^{2}+2x-15}+\frac{x^{2}+3x+9}{\left(x-3\right)\left(x^{2}+3x+9\right)}$$

Cancel out x^{2}+3x+9 in both numerator and denominator.

$$\frac{x+1}{x^{2}-9}-\frac{2x}{x^{2}+2x-15}+\frac{1}{x-3}$$

Factor x^{2}-9. Factor x^{2}+2x-15.

$$\frac{x+1}{\left(x-3\right)\left(x+3\right)}-\frac{2x}{\left(x-3\right)\left(x+5\right)}+\frac{1}{x-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right) and \left(x-3\right)\left(x+5\right) is \left(x-3\right)\left(x+3\right)\left(x+5\right). Multiply \frac{x+1}{\left(x-3\right)\left(x+3\right)} times \frac{x+5}{x+5}. Multiply \frac{2x}{\left(x-3\right)\left(x+5\right)} times \frac{x+3}{x+3}.

$$\frac{\left(x+1\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}-\frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{1}{x-3}$$

Since \frac{\left(x+1\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} and \frac{2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\left(x+5\right)-2x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{1}{x-3}$$

Do the multiplications in \left(x+1\right)\left(x+5\right)-2x\left(x+3\right).

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

Combine like terms in x^{2}+5x+x+5-2x^{2}-6x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-3\right)\left(x+3\right)\left(x+5\right) and x-3 is \left(x-3\right)\left(x+3\right)\left(x+5\right). Multiply \frac{1}{x-3} times \frac{\left(x+3\right)\left(x+5\right)}{\left(x+3\right)\left(x+5\right)}.

$$\frac{-x^{2}+5}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}+\frac{\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Since \frac{-x^{2}+5}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} and \frac{\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}+5+\left(x+3\right)\left(x+5\right)}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Do the multiplications in -x^{2}+5+\left(x+3\right)\left(x+5\right).

$$\frac{-x^{2}+5+x^{2}+5x+3x+15}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Combine like terms in -x^{2}+5+x^{2}+5x+3x+15.

$$\frac{20+8x}{\left(x-3\right)\left(x+3\right)\left(x+5\right)}$$

Expand \left(x-3\right)\left(x+3\right)\left(x+5\right).

$$\frac{20+8x}{x^{3}+5x^{2}-9x-45}$$