Add $a+8$ and $ \dfrac{a}{6a+5} $ to get $ \dfrac{ \color{purple}{ 6a^2+54a+40 } }{ 6a+5 }$.

Step 1: Write $ a+8 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 6a+5 }$.

$$ \begin{aligned} a+8+ \frac{a}{6a+5} & \xlongequal{\text{Step 1}} \frac{a+8}{\color{red}{1}} + \frac{a}{6a+5} = \frac{ \left( a+8 \right) \cdot \color{blue}{ \left( 6a+5 \right) }}{ 1 \cdot \color{blue}{ \left( 6a+5 \right) }} + \frac{ a }{ 6a+5 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6a^2+5a+48a+40 } }{ 6a+5 } + \frac{ \color{purple}{ a } }{ 6a+5 }=\frac{ \color{purple}{ 6a^2+54a+40 } }{ 6a+5 } \end{aligned} $$