Add $3x^2+16x$ and $ \dfrac{5}{x} $ to get $ \dfrac{ \color{purple}{ 3x^3+16x^2+5 } }{ x }$.

Step 1: Write $ 3x^2+16x $ 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 first fraction by $\color{blue}{ x }$.

$$ \begin{aligned} 3x^2+16x+ \frac{5}{x} & \xlongequal{\text{Step 1}} \frac{3x^2+16x}{\color{red}{1}} + \frac{5}{x} = \frac{ \left( 3x^2+16x \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 5 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3x^3+16x^2 } }{ x } + \frac{ \color{purple}{ 5 } }{ x }=\frac{ \color{purple}{ 3x^3+16x^2+5 } }{ x } \end{aligned} $$

Add $ \dfrac{3x^3+16x^2+5}{x} $ and $ 5 $ to get $ \dfrac{ \color{purple}{ 3x^3+16x^2+5x+5 } }{ x }$.

Step 1: Write $ 5 $ 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 $\color{blue}{x}$.

$$ \begin{aligned} \frac{3x^3+16x^2+5}{x} +5 & \xlongequal{\text{Step 1}} \frac{3x^3+16x^2+5}{x} + \frac{5}{\color{red}{1}} = \frac{ 3x^3+16x^2+5 }{ x } + \frac{ 5 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3x^3+16x^2+5 } }{ x } + \frac{ \color{purple}{ 5x } }{ x }=\frac{ \color{purple}{ 3x^3+16x^2+5x+5 } }{ x } \end{aligned} $$