Add $3x$ and $ \dfrac{1}{x^2} $ to get $ \dfrac{ \color{purple}{ 3x^3+1 } }{ x^2 }$.

Step 1: Write $ 3x $ 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 $\color{blue}{ x^2 }$.

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

Multiply $ \dfrac{1}{3} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $x$ from $ \dfrac{3x^3+1}{x^2} $ to get $ \dfrac{ \color{purple}{ 2x^3+1 } }{ x^2 }$.

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

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

Multiply $ \dfrac{1}{3} $ by $ x^2 $ to get $ \dfrac{ x^2 }{ 3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2x^3+1}{x^2} $ and $ \dfrac{x^2}{3} $ to get $ \dfrac{ \color{purple}{ x^4+6x^3+3 } }{ 3x^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}{ 3 }$ and the second by $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{2x^3+1}{x^2} + \frac{x^2}{3} & = \frac{ \left( 2x^3+1 \right) \cdot \color{blue}{ 3 }}{ x^2 \cdot \color{blue}{ 3 }} + \frac{ x^2 \cdot \color{blue}{ x^2 }}{ 3 \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 6x^3+3 } }{ 3x^2 } + \frac{ \color{purple}{ x^4 } }{ 3x^2 }=\frac{ \color{purple}{ x^4+6x^3+3 } }{ 3x^2 } \end{aligned} $$

Add $ \dfrac{x^4+6x^3+3}{3x^2} $ and $ x $ to get $ \dfrac{ \color{purple}{ x^4+9x^3+3 } }{ 3x^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 second fraction by $\color{blue}{3x^2}$.

$$ \begin{aligned} \frac{x^4+6x^3+3}{3x^2} +x & \xlongequal{\text{Step 1}} \frac{x^4+6x^3+3}{3x^2} + \frac{x}{\color{red}{1}} = \frac{ x^4+6x^3+3 }{ 3x^2 } + \frac{ x \cdot \color{blue}{ 3x^2 }}{ 1 \cdot \color{blue}{ 3x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+6x^3+3 } }{ 3x^2 } + \frac{ \color{purple}{ 3x^3 } }{ 3x^2 }=\frac{ \color{purple}{ x^4+9x^3+3 } }{ 3x^2 } \end{aligned} $$