Multiply $4$ by $ \dfrac{x^2}{x} $ to get $ \dfrac{ 4x^2 }{ x } $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{x^3}{x^2} $ and $ \dfrac{4x^2}{x} $ to get $ \dfrac{ \color{purple}{ 5x^4 } }{ x^3 }$.

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 }$ and the second by $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{x^3}{x^2} + \frac{4x^2}{x} & = \frac{ x^3 \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} + \frac{ 4x^2 \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ x^4 } }{ x^3 } + \frac{ \color{purple}{ 4x^4 } }{ x^3 }=\frac{ \color{purple}{ 5x^4 } }{ x^3 } \end{aligned} $$

Subtract $12x$ from $ \dfrac{5x^4}{x^3} $ to get $ \dfrac{ \color{purple}{ -7x^4 } }{ x^3 }$.

Step 1: Write $ 12x $ 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 $\color{blue}{x^3}$.

$$ \begin{aligned} \frac{5x^4}{x^3} -12x & \xlongequal{\text{Step 1}} \frac{5x^4}{x^3} - \frac{12x}{\color{red}{1}} = \frac{ 5x^4 }{ x^3 } - \frac{ 12x \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5x^4 } }{ x^3 } - \frac{ \color{purple}{ 12x^4 } }{ x^3 }=\frac{ \color{purple}{ -7x^4 } }{ x^3 } \end{aligned} $$