Multiply $4x^2+4x+6$ by $ \dfrac{4x^2+2x+6}{2x} $ to get $ \dfrac{16x^4+24x^3+56x^2+36x+36}{2x} $.

Step 1: Write $ 4x^2+4x+6 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4x^2+4x+6 \cdot \frac{4x^2+2x+6}{2x} & \xlongequal{\text{Step 1}} \frac{4x^2+4x+6}{\color{red}{1}} \cdot \frac{4x^2+2x+6}{2x} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 4x^2+4x+6 \right) \cdot \left( 4x^2+2x+6 \right) }{ 1 \cdot 2x } \xlongequal{\text{Step 3}} \frac{ 16x^4+8x^3+24x^2+16x^3+8x^2+24x+24x^2+12x+36 }{ 2x } = \\[1ex] &= \frac{16x^4+24x^3+56x^2+36x+36}{2x} \end{aligned} $$

Add $ \dfrac{16x^4+24x^3+56x^2+36x+36}{2x} $ and $ 2x $ to get $ \dfrac{ \color{purple}{ 16x^4+24x^3+60x^2+36x+36 } }{ 2x }$.

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

$$ \begin{aligned} \frac{16x^4+24x^3+56x^2+36x+36}{2x} +2x & \xlongequal{\text{Step 1}} \frac{16x^4+24x^3+56x^2+36x+36}{2x} + \frac{2x}{\color{red}{1}} = \\[1ex] &= \frac{ 16x^4+24x^3+56x^2+36x+36 }{ 2x } + \frac{ 2x \cdot \color{blue}{ 2x }}{ 1 \cdot \color{blue}{ 2x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 16x^4+24x^3+56x^2+36x+36 } }{ 2x } + \frac{ \color{purple}{ 4x^2 } }{ 2x }=\frac{ \color{purple}{ 16x^4+24x^3+60x^2+36x+36 } }{ 2x } \end{aligned} $$

Subtract $ \dfrac{18}{x} $ from $ \dfrac{16x^4+24x^3+60x^2+36x+36}{2x} $ to get $ \dfrac{ \color{purple}{ 16x^5+24x^4+60x^3+36x^2 } }{ 2x^2 }$.

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 $\color{blue}{ 2x }$.

$$ \begin{aligned} \frac{16x^4+24x^3+60x^2+36x+36}{2x} - \frac{18}{x} & = \frac{ \left( 16x^4+24x^3+60x^2+36x+36 \right) \cdot \color{blue}{ x }}{ 2x \cdot \color{blue}{ x }} - \frac{ 18 \cdot \color{blue}{ 2x }}{ x \cdot \color{blue}{ 2x }} = \\[1ex] &=\frac{ \color{purple}{ 16x^5+24x^4+60x^3+36x^2+36x } }{ 2x^2 } - \frac{ \color{purple}{ 36x } }{ 2x^2 } = \\[1ex] &=\frac{ \color{purple}{ 16x^5+24x^4+60x^3+36x^2 } }{ 2x^2 } \end{aligned} $$