Add $x^2+6x$ and $ \dfrac{9}{x^3} $ to get $ \dfrac{ \color{purple}{ x^5+6x^4+9 } }{ x^3 }$.

Step 1: Write $ x^2+6x $ 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 $\color{blue}{ x^3 }$.

$$ \begin{aligned} x^2+6x+ \frac{9}{x^3} & \xlongequal{\text{Step 1}} \frac{x^2+6x}{\color{red}{1}} + \frac{9}{x^3} = \frac{ \left( x^2+6x \right) \cdot \color{blue}{ x^3 }}{ 1 \cdot \color{blue}{ x^3 }} + \frac{ 9 }{ x^3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^5+6x^4 } }{ x^3 } + \frac{ \color{purple}{ 9 } }{ x^3 }=\frac{ \color{purple}{ x^5+6x^4+9 } }{ x^3 } \end{aligned} $$

Multiply $27$ by $ \dfrac{1}{x} $ to get $ \dfrac{ 27 }{ x } $.

Step 1: Write $ 27 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 27 \cdot \frac{1}{x} & \xlongequal{\text{Step 1}} \frac{27}{\color{red}{1}} \cdot \frac{1}{x} \xlongequal{\text{Step 2}} \frac{ 27 \cdot 1 }{ 1 \cdot x } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 27 }{ x } \end{aligned} $$

Add $ \dfrac{x^5+6x^4+9}{x^3} $ and $ \dfrac{27}{x} $ to get $ \dfrac{ \color{purple}{ x^6+6x^5+27x^3+9x } }{ x^4 }$.

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

$$ \begin{aligned} \frac{x^5+6x^4+9}{x^3} + \frac{27}{x} & = \frac{ \left( x^5+6x^4+9 \right) \cdot \color{blue}{ x }}{ x^3 \cdot \color{blue}{ x }} + \frac{ 27 \cdot \color{blue}{ x^3 }}{ x \cdot \color{blue}{ x^3 }} = \\[1ex] &=\frac{ \color{purple}{ x^6+6x^5+9x } }{ x^4 } + \frac{ \color{purple}{ 27x^3 } }{ x^4 }=\frac{ \color{purple}{ x^6+6x^5+27x^3+9x } }{ x^4 } \end{aligned} $$

Add $ \dfrac{x^6+6x^5+27x^3+9x}{x^4} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ x^6+6x^5+3x^4+27x^3+9x } }{ 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 $\color{blue}{x^4}$.

$$ \begin{aligned} \frac{x^6+6x^5+27x^3+9x}{x^4} +3 & \xlongequal{\text{Step 1}} \frac{x^6+6x^5+27x^3+9x}{x^4} + \frac{3}{\color{red}{1}} = \frac{ x^6+6x^5+27x^3+9x }{ x^4 } + \frac{ 3 \cdot \color{blue}{ x^4 }}{ 1 \cdot \color{blue}{ x^4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^6+6x^5+27x^3+9x } }{ x^4 } + \frac{ \color{purple}{ 3x^4 } }{ x^4 }=\frac{ \color{purple}{ x^6+6x^5+3x^4+27x^3+9x } }{ x^4 } \end{aligned} $$