Calculate: /(2) x+1+/(3) x+2 ?

Expression: $$\frac { 2 } { x + 1 } + \frac { 3 } { x + 2 } ?$$

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 $\frac{2}{x+1}$ times $\frac{x+2}{x+2}$. Multiply $\frac{3}{x+2}$ times $\frac{x+1}{x+1}$.

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

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

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

Do the multiplications in $2\left(x+2\right)+3\left(x+1\right)$.

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

Combine like terms in $2x+4+3x+3$.

$$\frac{5x+7}{\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{5x+7}{x^{2}+3x+2}$$