Add $6x^4$ and $ \dfrac{7}{x^2} $ to get $ \dfrac{ \color{purple}{ 6x^6+7 } }{ x^2 }$.

Step 1: Write $ 6x^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 first fraction by $\color{blue}{ x^2 }$.

$$ \begin{aligned} 6x^4+ \frac{7}{x^2} & \xlongequal{\text{Step 1}} \frac{6x^4}{\color{red}{1}} + \frac{7}{x^2} = \frac{ 6x^4 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 7 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6x^6 } }{ x^2 } + \frac{ \color{purple}{ 7 } }{ x^2 }=\frac{ \color{purple}{ 6x^6+7 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{6x^6+7}{x^2} $ and $ 9x $ to get $ \dfrac{ \color{purple}{ 6x^6+9x^3+7 } }{ x^2 }$.

Step 1: Write $ 9x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{6x^6+7}{x^2} +9x & \xlongequal{\text{Step 1}} \frac{6x^6+7}{x^2} + \frac{9x}{\color{red}{1}} = \frac{ 6x^6+7 }{ x^2 } + \frac{ 9x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6x^6+7 } }{ x^2 } + \frac{ \color{purple}{ 9x^3 } }{ x^2 }=\frac{ \color{purple}{ 6x^6+9x^3+7 } }{ x^2 } \end{aligned} $$