Add $ \dfrac{m+7}{m^2} $ and $ 4m $ to get $ \dfrac{ \color{purple}{ 4m^3+m+7 } }{ m^2 }$.

Step 1: Write $ 4m $ 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 second fraction by $\color{blue}{m^2}$.

$$ \begin{aligned} \frac{m+7}{m^2} +4m & \xlongequal{\text{Step 1}} \frac{m+7}{m^2} + \frac{4m}{\color{red}{1}} = \frac{ m+7 }{ m^2 } + \frac{ 4m \cdot \color{blue}{ m^2 }}{ 1 \cdot \color{blue}{ m^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ m+7 } }{ m^2 } + \frac{ \color{purple}{ 4m^3 } }{ m^2 }=\frac{ \color{purple}{ 4m^3+m+7 } }{ m^2 } \end{aligned} $$

Subtract $21$ from $ \dfrac{4m^3+m+7}{m^2} $ to get $ \dfrac{ \color{purple}{ 4m^3-21m^2+m+7 } }{ m^2 }$.

Step 1: Write $ 21 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{m^2}$.

$$ \begin{aligned} \frac{4m^3+m+7}{m^2} -21 & \xlongequal{\text{Step 1}} \frac{4m^3+m+7}{m^2} - \frac{21}{\color{red}{1}} = \frac{ 4m^3+m+7 }{ m^2 } - \frac{ 21 \cdot \color{blue}{ m^2 }}{ 1 \cdot \color{blue}{ m^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4m^3+m+7 } }{ m^2 } - \frac{ \color{purple}{ 21m^2 } }{ m^2 }=\frac{ \color{purple}{ 4m^3-21m^2+m+7 } }{ m^2 } \end{aligned} $$