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

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

Evaluate

$$\frac{x^{3}+x^{2}-15x-14}{\left(x+1\right)\left(x+2\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in 12-x^{2}-2x+2x+4.

$$-\frac{16-x^{2}}{x+2}+\frac{1}{x+1}$$

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{16-x^{2}}{x+2} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in -16x-16+x^{3}+x^{2}+x+2.

$$\frac{-15x-14+x^{3}+x^{2}}{\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-15x-14+x^{3}+x^{2}}{x^{2}+3x+2}$$

Expand

$$\frac{x^{3}+x^{2}-15x-14}{\left(x+1\right)\left(x+2\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in 12-x^{2}-2x+2x+4.

$$-\frac{16-x^{2}}{x+2}+\frac{1}{x+1}$$

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{16-x^{2}}{x+2} times \frac{x+1}{x+1}. Multiply \frac{1}{x+1} times \frac{x+2}{x+2}.

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

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

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

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

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

Combine like terms in -16x-16+x^{3}+x^{2}+x+2.

$$\frac{-15x-14+x^{3}+x^{2}}{\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{-15x-14+x^{3}+x^{2}}{x^{2}+3x+2}$$