Add $ \dfrac{16}{x^2} $ and $ \dfrac{4}{x} $ to get $ \dfrac{ \color{purple}{ 4x^2+16x } }{ x^3 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x }$ and the second by $\color{blue}{ x^2 }$.

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

Divide $ \dfrac{x^2}{x+4} $ by $ \dfrac{4x^2+16x}{x^3} $ to get $ \dfrac{x^5}{4x^3+32x^2+64x} $.

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.

$$ \begin{aligned} \frac{ \frac{x^2}{x+4} }{ \frac{\color{blue}{4x^2+16x}}{\color{blue}{x^3}} } & \xlongequal{\text{Step 1}} \frac{x^2}{x+4} \cdot \frac{\color{blue}{x^3}}{\color{blue}{4x^2+16x}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ x^2 \cdot x^3 }{ \left( x+4 \right) \cdot \left( 4x^2+16x \right) } \xlongequal{\text{Step 3}} \frac{ x^5 }{ 4x^3+16x^2+16x^2+64x } = \\[1ex] &= \frac{x^5}{4x^3+32x^2+64x} \end{aligned} $$