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

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 $\color{blue}{ x }$.

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

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

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

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

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

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

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

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

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

Subtract $10$ from $ \dfrac{4x^3+2}{x^2} $ to get $ \dfrac{ \color{purple}{ 4x^3-10x^2+2 } }{ x^2 }$.

Step 1: Write $ 10 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{4x^3+2}{x^2} -10 & \xlongequal{\text{Step 1}} \frac{4x^3+2}{x^2} - \frac{10}{\color{red}{1}} = \frac{ 4x^3+2 }{ x^2 } - \frac{ 10 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^3+2 } }{ x^2 } - \frac{ \color{purple}{ 10x^2 } }{ x^2 }=\frac{ \color{purple}{ 4x^3-10x^2+2 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{x^2+5x+3}{x} $ and $ \dfrac{4x^3-10x^2+2}{x^2} $ to get $ \dfrac{ \color{purple}{ 5x^4-5x^3+3x^2+2x } }{ 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^2 }$ and the second by $\color{blue}{ x }$.

$$ \begin{aligned} \frac{x^2+5x+3}{x} + \frac{4x^3-10x^2+2}{x^2} & = \frac{ \left( x^2+5x+3 \right) \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} + \frac{ \left( 4x^3-10x^2+2 \right) \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ x^4+5x^3+3x^2 } }{ x^3 } + \frac{ \color{purple}{ 4x^4-10x^3+2x } }{ x^3 }=\frac{ \color{purple}{ 5x^4-5x^3+3x^2+2x } }{ x^3 } \end{aligned} $$