Add $ \dfrac{6}{m+3} $ and $ \dfrac{4}{m+5} $ to get $ \dfrac{ \color{purple}{ 10m+42 } }{ m^2+8m+15 }$.

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}{ m+5 }$ and the second by $\color{blue}{ m+3 }$.

$$ \begin{aligned} \frac{6}{m+3} + \frac{4}{m+5} & = \frac{ 6 \cdot \color{blue}{ \left( m+5 \right) }}{ \left( m+3 \right) \cdot \color{blue}{ \left( m+5 \right) }} + \frac{ 4 \cdot \color{blue}{ \left( m+3 \right) }}{ \left( m+5 \right) \cdot \color{blue}{ \left( m+3 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 6m+30 } }{ m^2+5m+3m+15 } + \frac{ \color{purple}{ 4m+12 } }{ m^2+5m+3m+15 }=\frac{ \color{purple}{ 10m+42 } }{ m^2+8m+15 } \end{aligned} $$

Subtract $ \dfrac{m+5}{6} $ from $ \dfrac{m+5}{m+3} $ to get $ \dfrac{ \color{purple}{ -m^2-2m+15 } }{ 6m+18 }$.

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}{ 6 }$ and the second by $\color{blue}{ m+3 }$.

$$ \begin{aligned} \frac{m+5}{m+3} - \frac{m+5}{6} & = \frac{ \left( m+5 \right) \cdot \color{blue}{ 6 }}{ \left( m+3 \right) \cdot \color{blue}{ 6 }} - \frac{ \left( m+5 \right) \cdot \color{blue}{ \left( m+3 \right) }}{ 6 \cdot \color{blue}{ \left( m+3 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 6m+30 } }{ 6m+18 } - \frac{ \color{purple}{ m^2+3m+5m+15 } }{ 6m+18 }=\frac{ \color{purple}{ 6m+30 - \left( m^2+3m+5m+15 \right) } }{ 6m+18 } = \\[1ex] &=\frac{ \color{purple}{ -m^2-2m+15 } }{ 6m+18 } \end{aligned} $$

Divide $ \dfrac{10m+42}{m^2+8m+15} $ by $ \dfrac{-m^2-2m+15}{6m+18} $ to get $ \dfrac{60m+252}{-m^3-7m^2+5m+75} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Factor numerators and denominators.

Step 3: Cancel common factors.

Step 4: Multiply numerators and denominators.

Step 5: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{10m+42}{m^2+8m+15} }{ \frac{\color{blue}{-m^2-2m+15}}{\color{blue}{6m+18}} } & \xlongequal{\text{Step 1}} \frac{10m+42}{m^2+8m+15} \cdot \frac{\color{blue}{6m+18}}{\color{blue}{-m^2-2m+15}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 10m+42 }{ \left( m+5 \right) \cdot \color{red}{ \left( m+3 \right) } } \cdot \frac{ 6 \cdot \color{red}{ \left( m+3 \right) } }{ -m^2-2m+15 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 10m+42 }{ m+5 } \cdot \frac{ 6 }{ -m^2-2m+15 } \xlongequal{\text{Step 4}} \frac{ \left( 10m+42 \right) \cdot 6 }{ \left( m+5 \right) \cdot \left( -m^2-2m+15 \right) } = \\[1ex] & \xlongequal{\text{Step 5}} \frac{ 60m+252 }{ -m^3-2m^2+15m-5m^2-10m+75 } = \frac{60m+252}{-m^3-7m^2+5m+75} \end{aligned} $$