Divide $ \dfrac{3x}{x^2-81} $ by $ 4x-16 $ to get $ \dfrac{ 3x }{ 4x^3-16x^2-324x+1296 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{3x}{x^2-81} }{4x-16} & \xlongequal{\text{Step 1}} \frac{3x}{x^2-81} \cdot \frac{\color{blue}{1}}{\color{blue}{4x-16}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3x \cdot 1 }{ \left( x^2-81 \right) \cdot \left( 4x-16 \right) } \xlongequal{\text{Step 3}} \frac{ 3x }{ 4x^3-16x^2-324x+1296 } \end{aligned} $$

Divide $ \dfrac{3x}{4x^3-16x^2-324x+1296} $ by $ 9x+81 $ to get $ \dfrac{3x}{36x^4+180x^3-4212x^2-14580x+104976} $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{3x}{4x^3-16x^2-324x+1296} }{9x+81} & \xlongequal{\text{Step 1}} \frac{3x}{4x^3-16x^2-324x+1296} \cdot \frac{\color{blue}{1}}{\color{blue}{9x+81}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 3x \cdot 1 }{ \left( 4x^3-16x^2-324x+1296 \right) \cdot \left( 9x+81 \right) } \xlongequal{\text{Step 3}} \frac{ 3x }{ 36x^4+324x^3-144x^3-1296x^2-2916x^2-26244x+11664x+104976 } = \\[1ex] &= \frac{3x}{36x^4+180x^3-4212x^2-14580x+104976} \end{aligned} $$