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

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

Multiply $ \dfrac{x^4+3ax^2-27a-81}{x^4} $ by $ \dfrac{x^{10}}{x^3} $ to get $ \dfrac{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} }{ x^7 } $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^4+3ax^2-27a-81}{x^4} \cdot \frac{x^{10}}{x^3} & \xlongequal{\text{Step 1}} \frac{ \left( x^4+3ax^2-27a-81 \right) \cdot x^{10} }{ x^4 \cdot x^3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} }{ x^7 } \end{aligned} $$

Subtract $81$ from $ \dfrac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^7} $ to get $ \dfrac{ \color{purple}{ x^{14}+3ax^{12}-27ax^{10}-81x^{10}-81x^7 } }{ x^7 }$.

Step 1: Write $ 81 $ 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 $\color{blue}{x^7}$.

$$ \begin{aligned} \frac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^7} -81 & \xlongequal{\text{Step 1}} \frac{x^{14}+3ax^{12}-27ax^{10}-81x^{10}}{x^7} - \frac{81}{\color{red}{1}} = \\[1ex] &= \frac{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} }{ x^7 } - \frac{ 81 \cdot \color{blue}{ x^7 }}{ 1 \cdot \color{blue}{ x^7 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^{14}+3ax^{12}-27ax^{10}-81x^{10} } }{ x^7 } - \frac{ \color{purple}{ 81x^7 } }{ x^7 } = \\[1ex] &=\frac{ \color{purple}{ x^{14}+3ax^{12}-27ax^{10}-81x^{10}-81x^7 } }{ x^7 } \end{aligned} $$