Solve {l}{-2x/x-1}{-x-1/x+2}{+3/x-1}{-4/x+1}

$$\left. \begin{array} { l } { - \frac { 2 x } { x - 1 } } \\ { - \frac { x - 1 } { x + 2 } } \\ { + \frac { 3 } { x - 1 } } \\ { - \frac { 4 } { x + 1 } } \end{array} \right.$$

Evaluate

$$\frac{13+2x-6x^{2}-3x^{3}}{\left(x+2\right)\left(x^{2}-1\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in -2x^{2}-4x-x^{2}+x+x-1.

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

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+2\right). Multiply \frac{3}{x-1} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in -3x^{2}-2x-1+3x+6.

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

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{-3x^{2}+x+5}{\left(x-1\right)\left(x+2\right)} times \frac{x+1}{x+1}. Multiply \frac{4}{x+1} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

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

Since \frac{\left(-3x^{2}+x+5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{4\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.

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

Do the multiplications in \left(-3x^{2}+x+5\right)\left(x+1\right)-4\left(x-1\right)\left(x+2\right).

$$\frac{-3x^{3}-3x^{2}+x^{2}+x+5x+5-4x^{2}-8x+4x+8}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in -3x^{3}-3x^{2}+x^{2}+x+5x+5-4x^{2}-8x+4x+8.

$$\frac{-3x^{3}-6x^{2}+2x+13}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-3x^{3}-6x^{2}+2x+13}{x^{3}+2x^{2}-x-2}$$

Expand

$$\frac{13+2x-6x^{2}-3x^{3}}{\left(x+2\right)\left(x^{2}-1\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in -2x^{2}-4x-x^{2}+x+x-1.

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

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+2\right). Multiply \frac{3}{x-1} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in -3x^{2}-2x-1+3x+6.

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

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{-3x^{2}+x+5}{\left(x-1\right)\left(x+2\right)} times \frac{x+1}{x+1}. Multiply \frac{4}{x+1} times \frac{\left(x-1\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}.

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

Since \frac{\left(-3x^{2}+x+5\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{4\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.

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

Do the multiplications in \left(-3x^{2}+x+5\right)\left(x+1\right)-4\left(x-1\right)\left(x+2\right).

$$\frac{-3x^{3}-3x^{2}+x^{2}+x+5x+5-4x^{2}-8x+4x+8}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in -3x^{3}-3x^{2}+x^{2}+x+5x+5-4x^{2}-8x+4x+8.

$$\frac{-3x^{3}-6x^{2}+2x+13}{\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-3x^{3}-6x^{2}+2x+13}{x^{3}+2x^{2}-x-2}$$