Solve (x-2/x-1)-(2+12/x+4)

$$( \frac { x - 2 } { x - 1 } ) - ( 2 + \frac { 12 } { x + 4 } )$$

Evaluate

$$\frac{12-16x-x^{2}}{\left(x-1\right)\left(x+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x+4}{x+4}.

$$\frac{x-2}{x-1}-\left(\frac{2\left(x+4\right)}{x+4}+\frac{12}{x+4}\right)$$

Since \frac{2\left(x+4\right)}{x+4} and \frac{12}{x+4} have the same denominator, add them by adding their numerators.

$$\frac{x-2}{x-1}-\frac{2\left(x+4\right)+12}{x+4}$$

Do the multiplications in 2\left(x+4\right)+12.

$$\frac{x-2}{x-1}-\frac{2x+8+12}{x+4}$$

Combine like terms in 2x+8+12.

$$\frac{x-2}{x-1}-\frac{2x+20}{x+4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+4 is \left(x-1\right)\left(x+4\right). Multiply \frac{x-2}{x-1} times \frac{x+4}{x+4}. Multiply \frac{2x+20}{x+4} times \frac{x-1}{x-1}.

$$\frac{\left(x-2\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)}-\frac{\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)}$$

Since \frac{\left(x-2\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)} and \frac{\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(x+4\right)-\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)}$$

Do the multiplications in \left(x-2\right)\left(x+4\right)-\left(2x+20\right)\left(x-1\right).

$$\frac{x^{2}+4x-2x-8-2x^{2}+2x-20x+20}{\left(x-1\right)\left(x+4\right)}$$

Combine like terms in x^{2}+4x-2x-8-2x^{2}+2x-20x+20.

$$\frac{-x^{2}-16x+12}{\left(x-1\right)\left(x+4\right)}$$

Expand \left(x-1\right)\left(x+4\right).

$$\frac{-x^{2}-16x+12}{x^{2}+3x-4}$$

Expand

$$\frac{12-16x-x^{2}}{\left(x-1\right)\left(x+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x+4}{x+4}.

$$\frac{x-2}{x-1}-\left(\frac{2\left(x+4\right)}{x+4}+\frac{12}{x+4}\right)$$

Since \frac{2\left(x+4\right)}{x+4} and \frac{12}{x+4} have the same denominator, add them by adding their numerators.

$$\frac{x-2}{x-1}-\frac{2\left(x+4\right)+12}{x+4}$$

Do the multiplications in 2\left(x+4\right)+12.

$$\frac{x-2}{x-1}-\frac{2x+8+12}{x+4}$$

Combine like terms in 2x+8+12.

$$\frac{x-2}{x-1}-\frac{2x+20}{x+4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and x+4 is \left(x-1\right)\left(x+4\right). Multiply \frac{x-2}{x-1} times \frac{x+4}{x+4}. Multiply \frac{2x+20}{x+4} times \frac{x-1}{x-1}.

$$\frac{\left(x-2\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)}-\frac{\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)}$$

Since \frac{\left(x-2\right)\left(x+4\right)}{\left(x-1\right)\left(x+4\right)} and \frac{\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-2\right)\left(x+4\right)-\left(2x+20\right)\left(x-1\right)}{\left(x-1\right)\left(x+4\right)}$$

Do the multiplications in \left(x-2\right)\left(x+4\right)-\left(2x+20\right)\left(x-1\right).

$$\frac{x^{2}+4x-2x-8-2x^{2}+2x-20x+20}{\left(x-1\right)\left(x+4\right)}$$

Combine like terms in x^{2}+4x-2x-8-2x^{2}+2x-20x+20.

$$\frac{-x^{2}-16x+12}{\left(x-1\right)\left(x+4\right)}$$

Expand \left(x-1\right)\left(x+4\right).

$$\frac{-x^{2}-16x+12}{x^{2}+3x-4}$$