To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a^{2}+2a+4$ and $a-2$ is $\left(a-2\right)\left(a^{2}+2a+4\right)$. Multiply $\frac{a+2}{a^{2}+2a+4}$ times $\frac{a-2}{a-2}$. Multiply $\frac{1}{a-2}$ times $\frac{a^{2}+2a+4}{a^{2}+2a+4}$.
Since $\frac{\left(a+2\right)\left(a-2\right)}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ and $\frac{a^{2}+2a+4}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ have the same denominator, subtract them by subtracting their numerators.
Since $\frac{-2a-8}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ and $\frac{a^{3}+2a}{\left(a-2\right)\left(a^{2}+2a+4\right)}$ have the same denominator, add them by adding their numerators.
