Add $ \dfrac{4}{x^2} $ and $ 7x $ to get $ \dfrac{ \color{purple}{ 7x^3+4 } }{ x^2 }$.

Step 1: Write $ 7x $ 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 $\color{blue}{x^2}$.

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

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

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{7x^3+4}{x^2} $ and $ 12 $ to get $ \dfrac{ \color{purple}{ 7x^3+12x^2+4 } }{ x^2 }$.

Step 1: Write $ 12 $ 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 $\color{blue}{x^2}$.

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

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

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{7x^3+12x^2+4}{x^2} $ and $ \dfrac{3x}{x} $ to get $ \dfrac{ \color{purple}{ 7x^4+15x^3+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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} \frac{7x^3+12x^2+4}{x^2} + \frac{3x}{x} & = \frac{ \left( 7x^3+12x^2+4 \right) \cdot \color{blue}{ x }}{ x^2 \cdot \color{blue}{ x }} + \frac{ 3x \cdot \color{blue}{ x^2 }}{ x \cdot \color{blue}{ x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 7x^4+12x^3+4x } }{ x^3 } + \frac{ \color{purple}{ 3x^3 } }{ x^3 }=\frac{ \color{purple}{ 7x^4+15x^3+4x } }{ x^3 } \end{aligned} $$

Add $ \dfrac{7x^4+15x^3+4x}{x^3} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ 7x^4+18x^3+4x } }{ x^3 }$.

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 $\color{blue}{x^3}$.

$$ \begin{aligned} \frac{7x^4+15x^3+4x}{x^3} +3 & \xlongequal{\text{Step 1}} \frac{7x^4+15x^3+4x}{x^3} + \frac{3}{\color{red}{1}} = \frac{ 7x^4+15x^3+4x }{ x^3 } + \frac{ 3 \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x^4+15x^3+4x } }{ x^3 } + \frac{ \color{purple}{ 3x^3 } }{ x^3 }=\frac{ \color{purple}{ 7x^4+18x^3+4x } }{ x^3 } \end{aligned} $$

Subtract $ \dfrac{2}{x} $ from $ \dfrac{7x^4+18x^3+4x}{x^3} $ to get $ \dfrac{ \color{purple}{ 7x^5+18x^4-2x^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 $\color{blue}{ x^3 }$.

$$ \begin{aligned} \frac{7x^4+18x^3+4x}{x^3} - \frac{2}{x} & = \frac{ \left( 7x^4+18x^3+4x \right) \cdot \color{blue}{ x }}{ x^3 \cdot \color{blue}{ x }} - \frac{ 2 \cdot \color{blue}{ x^3 }}{ x \cdot \color{blue}{ x^3 }} = \\[1ex] &=\frac{ \color{purple}{ 7x^5+18x^4+4x^2 } }{ x^4 } - \frac{ \color{purple}{ 2x^3 } }{ x^4 }=\frac{ \color{purple}{ 7x^5+18x^4-2x^3+4x^2 } }{ x^4 } \end{aligned} $$

Add $ \dfrac{7x^5+18x^4-2x^3+4x^2}{x^4} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ 7x^5+22x^4-2x^3+4x^2 } }{ x^4 }$.

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

$$ \begin{aligned} \frac{7x^5+18x^4-2x^3+4x^2}{x^4} +4 & \xlongequal{\text{Step 1}} \frac{7x^5+18x^4-2x^3+4x^2}{x^4} + \frac{4}{\color{red}{1}} = \frac{ 7x^5+18x^4-2x^3+4x^2 }{ x^4 } + \frac{ 4 \cdot \color{blue}{ x^4 }}{ 1 \cdot \color{blue}{ x^4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 7x^5+18x^4-2x^3+4x^2 } }{ x^4 } + \frac{ \color{purple}{ 4x^4 } }{ x^4 }=\frac{ \color{purple}{ 7x^5+22x^4-2x^3+4x^2 } }{ x^4 } \end{aligned} $$