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

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

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

Multiply $ \dfrac{6}{x} $ by $ 2 $ to get $ \dfrac{ 12 }{ x } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{6}{x} \cdot 2 & \xlongequal{\text{Step 1}} \frac{6}{x} \cdot \frac{2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6 \cdot 2 }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 12 }{ x } \end{aligned} $$

Subtract $5x$ from $ \dfrac{x^3+4}{x^2} $ to get $ \dfrac{ \color{purple}{ -4x^3+4 } }{ x^2 }$.

Step 1: Write $ 5x $ 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 $\color{blue}{x^2}$.

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

Multiply $ \dfrac{6}{x} $ by $ 2 $ to get $ \dfrac{ 12 }{ x } $.

Step 1: Write $ 2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{6}{x} \cdot 2 & \xlongequal{\text{Step 1}} \frac{6}{x} \cdot \frac{2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6 \cdot 2 }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 12 }{ x } \end{aligned} $$

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

$$ \begin{aligned} \frac{-4x^3+4}{x^2} + \frac{12}{x} & = \frac{ \left( -4x^3+4 \right) \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} + \frac{ 12 \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ -4x^4+4x } }{ x^3 } + \frac{ \color{purple}{ 12x^2 } }{ x^3 }=\frac{ \color{purple}{ -4x^4+12x^2+4x } }{ x^3 } \end{aligned} $$

Subtract $ \dfrac{16}{x} $ from $ \dfrac{-4x^4+12x^2+4x}{x^3} $ to get $ \dfrac{ \color{purple}{ -4x^5-4x^3+4x^2 } }{ x^4 }$.

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}{ x }$ and the second by $\color{blue}{ x^3 }$.

$$ \begin{aligned} \frac{-4x^4+12x^2+4x}{x^3} - \frac{16}{x} & = \frac{ \left( -4x^4+12x^2+4x \right) \cdot \color{blue}{ x }}{ x^3 \cdot \color{blue}{ x }} - \frac{ 16 \cdot \color{blue}{ x^3 }}{ x \cdot \color{blue}{ x^3 }} = \\[1ex] &=\frac{ \color{purple}{ -4x^5+12x^3+4x^2 } }{ x^4 } - \frac{ \color{purple}{ 16x^3 } }{ x^4 }=\frac{ \color{purple}{ -4x^5-4x^3+4x^2 } }{ x^4 } \end{aligned} $$

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

Step 1: Write $ 3 $ 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 $\color{blue}{x^4}$.

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