Combine like terms:

$$ \, \color{blue}{ \cancel{x}} \,+ \color{green}{6x} -4x^2-16 \, \color{green}{ -\cancel{x}} \, = -4x^2+ \color{green}{6x} -16 $$

Subtract $ \dfrac{4}{x^2} $ from $ -4x^2+6x-16 $ to get $ \dfrac{ \color{purple}{ -4x^4+6x^3-16x^2-4 } }{ x^2 }$.

Step 1: Write $ -4x^2+6x-16 $ 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 first fraction by $\color{blue}{ x^2 }$.

$$ \begin{aligned} -4x^2+6x-16- \frac{4}{x^2} & \xlongequal{\text{Step 1}} \frac{-4x^2+6x-16}{\color{red}{1}} - \frac{4}{x^2} = \frac{ \left( -4x^2+6x-16 \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} - \frac{ 4 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4+6x^3-16x^2 } }{ x^2 } - \frac{ \color{purple}{ 4 } }{ x^2 }=\frac{ \color{purple}{ -4x^4+6x^3-16x^2-4 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{-4x^4+6x^3-16x^2-4}{x^2} $ and $ 2x $ to get $ \dfrac{ \color{purple}{ -4x^4+8x^3-16x^2-4 } }{ x^2 }$.

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

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

$$ \begin{aligned} \frac{-4x^4+6x^3-16x^2-4}{x^2} +2x & \xlongequal{\text{Step 1}} \frac{-4x^4+6x^3-16x^2-4}{x^2} + \frac{2x}{\color{red}{1}} = \frac{ -4x^4+6x^3-16x^2-4 }{ x^2 } + \frac{ 2x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4+6x^3-16x^2-4 } }{ x^2 } + \frac{ \color{purple}{ 2x^3 } }{ x^2 }=\frac{ \color{purple}{ -4x^4+8x^3-16x^2-4 } }{ x^2 } \end{aligned} $$

Subtract $2416x$ from $ \dfrac{-4x^4+8x^3-16x^2-4}{x^2} $ to get $ \dfrac{ \color{purple}{ -4x^4-2408x^3-16x^2-4 } }{ x^2 }$.

Step 1: Write $ 2416x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{-4x^4+8x^3-16x^2-4}{x^2} -2416x & \xlongequal{\text{Step 1}} \frac{-4x^4+8x^3-16x^2-4}{x^2} - \frac{2416x}{\color{red}{1}} = \\[1ex] &= \frac{ -4x^4+8x^3-16x^2-4 }{ x^2 } - \frac{ 2416x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4+8x^3-16x^2-4 } }{ x^2 } - \frac{ \color{purple}{ 2416x^3 } }{ x^2 }=\frac{ \color{purple}{ -4x^4-2408x^3-16x^2-4 } }{ x^2 } \end{aligned} $$

Subtract $x^3$ from $ \dfrac{-4x^4-2408x^3-16x^2-4}{x^2} $ to get $ \dfrac{ \color{purple}{ -x^5-4x^4-2408x^3-16x^2-4 } }{ x^2 }$.

Step 1: Write $ x^3 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{-4x^4-2408x^3-16x^2-4}{x^2} -x^3 & \xlongequal{\text{Step 1}} \frac{-4x^4-2408x^3-16x^2-4}{x^2} - \frac{x^3}{\color{red}{1}} = \\[1ex] &= \frac{ -4x^4-2408x^3-16x^2-4 }{ x^2 } - \frac{ x^3 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4-2408x^3-16x^2-4 } }{ x^2 } - \frac{ \color{purple}{ x^5 } }{ x^2 }=\frac{ \color{purple}{ -x^5-4x^4-2408x^3-16x^2-4 } }{ x^2 } \end{aligned} $$