Add $4x^2$ and $ \dfrac{12}{x} $ to get $ \dfrac{ \color{purple}{ 4x^3+12 } }{ x }$.

Step 1: Write $ 4x^2 $ 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 $\color{blue}{ x }$.

$$ \begin{aligned} 4x^2+ \frac{12}{x} & \xlongequal{\text{Step 1}} \frac{4x^2}{\color{red}{1}} + \frac{12}{x} = \frac{ 4x^2 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 12 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^3 } }{ x } + \frac{ \color{purple}{ 12 } }{ x }=\frac{ \color{purple}{ 4x^3+12 } }{ x } \end{aligned} $$

Add $ \dfrac{4x^3+12}{x} $ and $ 7 $ to get $ \dfrac{ \color{purple}{ 4x^3+7x+12 } }{ x }$.

Step 1: Write $ 7 $ 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 $\color{blue}{x}$.

$$ \begin{aligned} \frac{4x^3+12}{x} +7 & \xlongequal{\text{Step 1}} \frac{4x^3+12}{x} + \frac{7}{\color{red}{1}} = \frac{ 4x^3+12 }{ x } + \frac{ 7 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^3+12 } }{ x } + \frac{ \color{purple}{ 7x } }{ x }=\frac{ \color{purple}{ 4x^3+7x+12 } }{ x } \end{aligned} $$