Subtract $ \dfrac{4}{x^2} $ from $ \dfrac{3}{4} $ to get $ \dfrac{ \color{purple}{ 3x^2-16 } }{ 4x^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^2 }$ and the second by $\color{blue}{ 4 }$.

$$ \begin{aligned} \frac{3}{4} - \frac{4}{x^2} & = \frac{ 3 \cdot \color{blue}{ x^2 }}{ 4 \cdot \color{blue}{ x^2 }} - \frac{ 4 \cdot \color{blue}{ 4 }}{ x^2 \cdot \color{blue}{ 4 }} = \\[1ex] &=\frac{ \color{purple}{ 3x^2 } }{ 4x^2 } - \frac{ \color{purple}{ 16 } }{ 4x^2 }=\frac{ \color{purple}{ 3x^2-16 } }{ 4x^2 } \end{aligned} $$

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

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

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

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Multiply $4$ by $ \dfrac{4x^3}{3x^2-16} $ to get $ \dfrac{ 16x^3 }{ 3x^2-16 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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