Solve x+4/x+2-x+5/x+1-x-5/x-1+x-4/x-2

$$\frac { x + 4 } { x + 2 } - \frac { x + 5 } { x + 1 } - \frac { x - 5 } { x - 1 } + \frac { x - 4 } { x - 2 }$$

Evaluate

$$-\frac{24}{\left(x^{2}-4\right)\left(x^{2}-1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+1 is \left(x+1\right)\left(x+2\right). Multiply \frac{x+4}{x+2} times \frac{x+1}{x+1}. Multiply \frac{x+5}{x+1} times \frac{x+2}{x+2}.

$$\frac{\left(x+4\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}-\frac{\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Since \frac{\left(x+4\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)} and \frac{\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+4\right)\left(x+1\right)-\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Do the multiplications in \left(x+4\right)\left(x+1\right)-\left(x+5\right)\left(x+2\right).

$$\frac{x^{2}+x+4x+4-x^{2}-2x-5x-10}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Combine like terms in x^{2}+x+4x+4-x^{2}-2x-5x-10.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x+2\right) and x-1 is \left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{-2x-6}{\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{x-5}{x-1} times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

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

Since \frac{\left(-2x-6\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(-2x-6\right)\left(x-1\right)-\left(x-5\right)\left(x+1\right)\left(x+2\right).

$$\frac{-2x^{2}+2x-6x+6-x^{3}-3x^{2}-2x+5x^{2}+15x+10}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{x-4}{x-2}$$

Combine like terms in -2x^{2}+2x-6x+6-x^{3}-3x^{2}-2x+5x^{2}+15x+10.

$$\frac{9x+16-x^{3}}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{x-4}{x-2}$$

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)\left(x+2\right) and x-2 is \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{9x+16-x^{3}}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{x-4}{x-2} times \frac{\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(9x+16-x^{3}\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(9x+16-x^{3}\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x+16-x^{3}\right)\left(x-2\right)+\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(9x+16-x^{3}\right)\left(x-2\right)+\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right).

$$\frac{9x^{2}-18x+16x-32-x^{4}+2x^{3}+x^{4}+2x^{3}-x^{2}-2x-4x^{3}-8x^{2}+4x+8}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in 9x^{2}-18x+16x-32-x^{4}+2x^{3}+x^{4}+2x^{3}-x^{2}-2x-4x^{3}-8x^{2}+4x+8.

$$\frac{-24}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-24}{x^{4}-5x^{2}+4}$$

Expand

$$-\frac{24}{\left(x^{2}-4\right)\left(x^{2}-1\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x+1 is \left(x+1\right)\left(x+2\right). Multiply \frac{x+4}{x+2} times \frac{x+1}{x+1}. Multiply \frac{x+5}{x+1} times \frac{x+2}{x+2}.

$$\frac{\left(x+4\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}-\frac{\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Since \frac{\left(x+4\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)} and \frac{\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+4\right)\left(x+1\right)-\left(x+5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Do the multiplications in \left(x+4\right)\left(x+1\right)-\left(x+5\right)\left(x+2\right).

$$\frac{x^{2}+x+4x+4-x^{2}-2x-5x-10}{\left(x+1\right)\left(x+2\right)}-\frac{x-5}{x-1}+\frac{x-4}{x-2}$$

Combine like terms in x^{2}+x+4x+4-x^{2}-2x-5x-10.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+1\right)\left(x+2\right) and x-1 is \left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{-2x-6}{\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{x-5}{x-1} times \frac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}.

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

Since \frac{\left(-2x-6\right)\left(x-1\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(-2x-6\right)\left(x-1\right)-\left(x-5\right)\left(x+1\right)\left(x+2\right).

$$\frac{-2x^{2}+2x-6x+6-x^{3}-3x^{2}-2x+5x^{2}+15x+10}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{x-4}{x-2}$$

Combine like terms in -2x^{2}+2x-6x+6-x^{3}-3x^{2}-2x+5x^{2}+15x+10.

$$\frac{9x+16-x^{3}}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{x-4}{x-2}$$

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)\left(x+2\right) and x-2 is \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{9x+16-x^{3}}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{x-4}{x-2} times \frac{\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(9x+16-x^{3}\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}+\frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(9x+16-x^{3}\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x+16-x^{3}\right)\left(x-2\right)+\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(9x+16-x^{3}\right)\left(x-2\right)+\left(x-4\right)\left(x-1\right)\left(x+1\right)\left(x+2\right).

$$\frac{9x^{2}-18x+16x-32-x^{4}+2x^{3}+x^{4}+2x^{3}-x^{2}-2x-4x^{3}-8x^{2}+4x+8}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in 9x^{2}-18x+16x-32-x^{4}+2x^{3}+x^{4}+2x^{3}-x^{2}-2x-4x^{3}-8x^{2}+4x+8.

$$\frac{-24}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-24}{x^{4}-5x^{2}+4}$$