Multiply $8$ by $ \dfrac{t}{8t+4} $ to get $ \dfrac{ 8t }{ 8t+4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 8 \cdot \frac{t}{8t+4} & \xlongequal{\text{Step 1}} \frac{8}{\color{red}{1}} \cdot \frac{t}{8t+4} \xlongequal{\text{Step 2}} \frac{ 8 \cdot t }{ 1 \cdot \left( 8t+4 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8t }{ 8t+4 } \end{aligned} $$

Add $ \dfrac{8t}{8t+4} $ and $ \dfrac{t}{16t+8} $ to get $ \dfrac{ \color{purple}{ 17t } }{ 16t+8 }$.

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

$$ \begin{aligned} \frac{8t}{8t+4} + \frac{t}{16t+8} & = \frac{ 8t \cdot \color{blue}{ 2 }}{ \left( 8t+4 \right) \cdot \color{blue}{ 2 }} + \frac{ t }{ 16t+8 } = \\[1ex] &=\frac{ \color{purple}{ 16t } }{ 16t+8 } + \frac{ \color{purple}{ t } }{ 16t+8 }=\frac{ \color{purple}{ 17t } }{ 16t+8 } \end{aligned} $$