Multiply $x^2-9$ by $ \dfrac{x^2+3a+9}{4} $ to get $ \dfrac{x^4+3ax^2-27a-81}{4} $.

Step 1: Write $ x^2-9 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} x^2-9 \cdot \frac{x^2+3a+9}{4} & \xlongequal{\text{Step 1}} \frac{x^2-9}{\color{red}{1}} \cdot \frac{x^2+3a+9}{4} \xlongequal{\text{Step 2}} \frac{ \left( x^2-9 \right) \cdot \left( x^2+3a+9 \right) }{ 1 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^4+3ax^2+ \cancel{9x^2} -\cancel{9x^2}-27a-81 }{ 4 } = \frac{x^4+3ax^2-27a-81}{4} \end{aligned} $$