Multiply $2$ by $ \dfrac{m}{m+1} $ to get $ \dfrac{ 2m }{ m+1 } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2m}{m+1} $ and $ \dfrac{6}{4m+12} $ to get $ \dfrac{ \color{purple}{ 8m^2+30m+6 } }{ 4m^2+16m+12 }$.

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}{ 4m+12 }$ and the second by $\color{blue}{ m+1 }$.

$$ \begin{aligned} \frac{2m}{m+1} + \frac{6}{4m+12} & = \frac{ 2m \cdot \color{blue}{ \left( 4m+12 \right) }}{ \left( m+1 \right) \cdot \color{blue}{ \left( 4m+12 \right) }} + \frac{ 6 \cdot \color{blue}{ \left( m+1 \right) }}{ \left( 4m+12 \right) \cdot \color{blue}{ \left( m+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 8m^2+24m } }{ 4m^2+12m+4m+12 } + \frac{ \color{purple}{ 6m+6 } }{ 4m^2+12m+4m+12 } = \\[1ex] &=\frac{ \color{purple}{ 8m^2+30m+6 } }{ 4m^2+16m+12 } \end{aligned} $$