Multiply $2$ by $ \dfrac{x^2}{x^4} $ to get $ \dfrac{ 2x^2 }{ x^4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{2x^2}{x^4} $ and $ 9x^3 $ to get $ \dfrac{ \color{purple}{ 9x^7+2x^2 } }{ x^4 }$.

Step 1: Write $ 9x^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 $\color{blue}{x^4}$.

$$ \begin{aligned} \frac{2x^2}{x^4} +9x^3 & \xlongequal{\text{Step 1}} \frac{2x^2}{x^4} + \frac{9x^3}{\color{red}{1}} = \frac{ 2x^2 }{ x^4 } + \frac{ 9x^3 \cdot \color{blue}{ x^4 }}{ 1 \cdot \color{blue}{ x^4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2 } }{ x^4 } + \frac{ \color{purple}{ 9x^7 } }{ x^4 }=\frac{ \color{purple}{ 9x^7+2x^2 } }{ x^4 } \end{aligned} $$