Multiply $ \dfrac{4x^2-36}{2} $ by $ x^2 $ to get $ \dfrac{ 4x^4-36x^2 }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4x^2-36}{2} \cdot x^2 & \xlongequal{\text{Step 1}} \frac{4x^2-36}{2} \cdot \frac{x^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 4x^2-36 \right) \cdot x^2 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4x^4-36x^2 }{ 2 } \end{aligned} $$

Subtract $12x$ from $ \dfrac{4x^4-36x^2}{2} $ to get $ \dfrac{ \color{purple}{ 4x^4-36x^2-24x } }{ 2 }$.

Step 1: Write $ 12x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{4x^4-36x^2}{2} -12x & \xlongequal{\text{Step 1}} \frac{4x^4-36x^2}{2} - \frac{12x}{\color{red}{1}} = \frac{ 4x^4-36x^2 }{ 2 } - \frac{ 12x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^4-36x^2 } }{ 2 } - \frac{ \color{purple}{ 24x } }{ 2 }=\frac{ \color{purple}{ 4x^4-36x^2-24x } }{ 2 } \end{aligned} $$

Add $ \dfrac{4x^4-36x^2-24x}{2} $ and $ 18 $ to get $ \dfrac{ \color{purple}{ 4x^4-36x^2-24x+36 } }{ 2 }$.

Step 1: Write $ 18 $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{4x^4-36x^2-24x}{2} +18 & \xlongequal{\text{Step 1}} \frac{4x^4-36x^2-24x}{2} + \frac{18}{\color{red}{1}} = \frac{ 4x^4-36x^2-24x }{ 2 } + \frac{ 18 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x^4-36x^2-24x } }{ 2 } + \frac{ \color{purple}{ 36 } }{ 2 }=\frac{ \color{purple}{ 4x^4-36x^2-24x+36 } }{ 2 } \end{aligned} $$