Multiply $3$ by $ \dfrac{m}{m+1} $ to get $ \dfrac{ 3m }{ m+1 } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $2$ by $ \dfrac{m}{m-5} $ to get $ \dfrac{ 2m }{ m-5 } $.

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.

$$ \begin{aligned} 2 \cdot \frac{m}{m-5} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{m}{m-5} \xlongequal{\text{Step 2}} \frac{ 2 \cdot m }{ 1 \cdot \left( m-5 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2m }{ m-5 } \end{aligned} $$

Subtract $ \dfrac{2m}{m-5} $ from $ \dfrac{3m}{m+1} $ to get $ \dfrac{ \color{purple}{ m^2-17m } }{ m^2-4m-5 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ m-5 }$ and the second by $\color{blue}{ m+1 }$.

$$ \begin{aligned} \frac{3m}{m+1} - \frac{2m}{m-5} & = \frac{ 3m \cdot \color{blue}{ \left( m-5 \right) }}{ \left( m+1 \right) \cdot \color{blue}{ \left( m-5 \right) }} - \frac{ 2m \cdot \color{blue}{ \left( m+1 \right) }}{ \left( m-5 \right) \cdot \color{blue}{ \left( m+1 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 3m^2-15m } }{ m^2-5m+m-5 } - \frac{ \color{purple}{ 2m^2+2m } }{ m^2-5m+m-5 }=\frac{ \color{purple}{ 3m^2-15m - \left( 2m^2+2m \right) } }{ m^2-4m-5 } = \\[1ex] &=\frac{ \color{purple}{ m^2-17m } }{ m^2-4m-5 } \end{aligned} $$