Add $4x^2$ and $ \dfrac{1}{64x^2} $ to get $ \dfrac{ \color{purple}{ 256x^4+1 } }{ 64x^2 }$.

Step 1: Write $ 4x^2 $ 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 $\color{blue}{ 64x^2 }$.

$$ \begin{aligned} 4x^2+ \frac{1}{64x^2} & \xlongequal{\text{Step 1}} \frac{4x^2}{\color{red}{1}} + \frac{1}{64x^2} = \frac{ 4x^2 \cdot \color{blue}{ 64x^2 }}{ 1 \cdot \color{blue}{ 64x^2 }} + \frac{ 1 }{ 64x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 256x^4 } }{ 64x^2 } + \frac{ \color{purple}{ 1 } }{ 64x^2 }=\frac{ \color{purple}{ 256x^4+1 } }{ 64x^2 } \end{aligned} $$

Add $ \dfrac{256x^4+1}{64x^2} $ and $ \dfrac{1}{2} $ to get $ \dfrac{ \color{purple}{ 512x^4+64x^2+2 } }{ 128x^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}{ 2 }$ and the second by $\color{blue}{ 64x^2 }$.

$$ \begin{aligned} \frac{256x^4+1}{64x^2} + \frac{1}{2} & = \frac{ \left( 256x^4+1 \right) \cdot \color{blue}{ 2 }}{ 64x^2 \cdot \color{blue}{ 2 }} + \frac{ 1 \cdot \color{blue}{ 64x^2 }}{ 2 \cdot \color{blue}{ 64x^2 }} = \\[1ex] &=\frac{ \color{purple}{ 512x^4+2 } }{ 128x^2 } + \frac{ \color{purple}{ 64x^2 } }{ 128x^2 }=\frac{ \color{purple}{ 512x^4+64x^2+2 } }{ 128x^2 } \end{aligned} $$