Cancel out $8y$ in both numerator and denominator.
$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $y+3$ and $y^{2}+9$ is $\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{1}{y+3}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{y+3}{y+3}$.
Since $\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)\left(y^{2}+9\right)$ and $y-3$ is $\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}$ times $\frac{y-3}{y-3}$. Multiply $\frac{2}{y-3}$ times $\frac{\left(y+3\right)\left(y^{2}+9\right)}{\left(y+3\right)\left(y^{2}+9\right)}$.
Since $\frac{\left(y^{2}+27+6y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{2\left(y+3\right)\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.
Cancel out $8y$ in both numerator and denominator.
$$\frac{1}{y+3}+\frac{6}{y^{2}+9}-\frac{2}{y-3}$$
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $y+3$ and $y^{2}+9$ is $\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{1}{y+3}$ times $\frac{y^{2}+9}{y^{2}+9}$. Multiply $\frac{6}{y^{2}+9}$ times $\frac{y+3}{y+3}$.
Since $\frac{y^{2}+9}{\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{6\left(y+3\right)}{\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, add them by adding their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(y+3\right)\left(y^{2}+9\right)$ and $y-3$ is $\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)$. Multiply $\frac{y^{2}+27+6y}{\left(y+3\right)\left(y^{2}+9\right)}$ times $\frac{y-3}{y-3}$. Multiply $\frac{2}{y-3}$ times $\frac{\left(y+3\right)\left(y^{2}+9\right)}{\left(y+3\right)\left(y^{2}+9\right)}$.
Since $\frac{\left(y^{2}+27+6y\right)\left(y-3\right)}{\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)}$ and $\frac{2\left(y+3\right)\left(y^{2}+9\right)}{\left(y-3\right)\left(y+3\right)\left(y^{2}+9\right)}$ have the same denominator, subtract them by subtracting their numerators.
