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

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x-2 is \left(x-2\right)\left(x+1\right). Multiply \frac{2-x}{x+1} times \frac{x-2}{x-2}. Multiply \frac{x+8}{x-2} times \frac{x+1}{x+1}.

$$\frac{\left(2-x\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Since \frac{\left(2-x\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} and \frac{\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2-x\right)\left(x-2\right)+\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Do the multiplications in \left(2-x\right)\left(x-2\right)+\left(x+8\right)\left(x+1\right).

$$\frac{2x-4-x^{2}+2x+x^{2}+x+8x+8}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

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

$$\frac{13x+4}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Factor x^{2}-x-2.

$$\frac{13x+4}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{\left(x-2\right)\left(x+1\right)}$$

Since \frac{13x+4}{\left(x-2\right)\left(x+1\right)} and \frac{4x}{\left(x-2\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{13x+4-4x}{\left(x-2\right)\left(x+1\right)}$$

Combine like terms in 13x+4-4x.

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

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

$$\frac{9x+4}{x^{2}-x-2}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x-2 is \left(x-2\right)\left(x+1\right). Multiply \frac{2-x}{x+1} times \frac{x-2}{x-2}. Multiply \frac{x+8}{x-2} times \frac{x+1}{x+1}.

$$\frac{\left(2-x\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Since \frac{\left(2-x\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} and \frac{\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2-x\right)\left(x-2\right)+\left(x+8\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Do the multiplications in \left(2-x\right)\left(x-2\right)+\left(x+8\right)\left(x+1\right).

$$\frac{2x-4-x^{2}+2x+x^{2}+x+8x+8}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

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

$$\frac{13x+4}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{x^{2}-x-2}$$

Factor x^{2}-x-2.

$$\frac{13x+4}{\left(x-2\right)\left(x+1\right)}-\frac{4x}{\left(x-2\right)\left(x+1\right)}$$

Since \frac{13x+4}{\left(x-2\right)\left(x+1\right)} and \frac{4x}{\left(x-2\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{13x+4-4x}{\left(x-2\right)\left(x+1\right)}$$

Combine like terms in 13x+4-4x.

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

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

$$\frac{9x+4}{x^{2}-x-2}$$