Solve (x+2/x+5)-(x^2+4)

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

Evaluate

$$-\frac{x^{3}+5x^{2}+3x+18}{x+5}$$

Short Solution Steps

To find the opposite of x^{2}+4, find the opposite of each term.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}-4 times \frac{x+5}{x+5}.

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

Since \frac{x+2}{x+5} and \frac{\left(-x^{2}-4\right)\left(x+5\right)}{x+5} have the same denominator, add them by adding their numerators.

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

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

$$\frac{x+2-x^{3}-5x^{2}-4x-20}{x+5}$$

Combine like terms in x+2-x^{3}-5x^{2}-4x-20.

$$\frac{-3x-18-x^{3}-5x^{2}}{x+5}$$

Expand

$$-\frac{x^{3}+5x^{2}+3x+18}{x+5}$$

Short Solution Steps

To find the opposite of x^{2}+4, find the opposite of each term.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -x^{2}-4 times \frac{x+5}{x+5}.

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

Since \frac{x+2}{x+5} and \frac{\left(-x^{2}-4\right)\left(x+5\right)}{x+5} have the same denominator, add them by adding their numerators.

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

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

$$\frac{x+2-x^{3}-5x^{2}-4x-20}{x+5}$$

Combine like terms in x+2-x^{3}-5x^{2}-4x-20.

$$\frac{-3x-18-x^{3}-5x^{2}}{x+5}$$