Solve 2/x+2+x/x^2+5x+6

$$\frac { 2 } { x + 2 } + \frac { x } { x ^ { 2 } + 5 x + 6 }$$

Evaluate

$$\frac{3}{x+3}$$

Short Solution Steps

Factor x^{2}+5x+6.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and \left(x+2\right)\left(x+3\right) is \left(x+2\right)\left(x+3\right). Multiply \frac{2}{x+2} times \frac{x+3}{x+3}.

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

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

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

Do the multiplications in 2\left(x+3\right)+x.

$$\frac{2x+6+x}{\left(x+2\right)\left(x+3\right)}$$

Combine like terms in 2x+6+x.

$$\frac{3x+6}{\left(x+2\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{3x+6}{\left(x+2\right)\left(x+3\right)}.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{3}{x+3}$$

Differentiate w.r.t. x

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