Subtract $4x^2$ from $ \dfrac{1}{x^2} $ to get $ \dfrac{ \color{purple}{ -4x^4+1 } }{ x^2 }$.

Step 1: Write $ 4x^2 $ 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 $\color{blue}{x^2}$.

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

Add $ \dfrac{-4x^4+1}{x^2} $ and $ 6x $ to get $ \dfrac{ \color{purple}{ -4x^4+6x^3+1 } }{ x^2 }$.

Step 1: Write $ 6x $ 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 $\color{blue}{x^2}$.

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

Add $ \dfrac{-4x^4+6x^3+1}{x^2} $ and $ \dfrac{8}{x^2} $ to get $ \dfrac{-4x^4+6x^3+9}{x^2} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{-4x^4+6x^3+1}{x^2} + \frac{8}{x^2} & = \frac{-4x^4+6x^3+1}{\color{blue}{x^2}} + \frac{8}{\color{blue}{x^2}} = \\[1ex] &=\frac{ -4x^4+6x^3+1 + 8 }{ \color{blue}{ x^2 }}= \frac{-4x^4+6x^3+9}{x^2} \end{aligned} $$

Add $ \dfrac{-4x^4+6x^3+9}{x^2} $ and $ 5x $ to get $ \dfrac{ \color{purple}{ -4x^4+11x^3+9 } }{ x^2 }$.

Step 1: Write $ 5x $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{-4x^4+6x^3+9}{x^2} +5x & \xlongequal{\text{Step 1}} \frac{-4x^4+6x^3+9}{x^2} + \frac{5x}{\color{red}{1}} = \frac{ -4x^4+6x^3+9 }{ x^2 } + \frac{ 5x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4+6x^3+9 } }{ x^2 } + \frac{ \color{purple}{ 5x^3 } }{ x^2 }=\frac{ \color{purple}{ -4x^4+11x^3+9 } }{ x^2 } \end{aligned} $$

Add $ \dfrac{-4x^4+11x^3+9}{x^2} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ -4x^4+11x^3+4x^2+9 } }{ x^2 }$.

Step 1: Write $ 4 $ 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 $\color{blue}{x^2}$.

$$ \begin{aligned} \frac{-4x^4+11x^3+9}{x^2} +4 & \xlongequal{\text{Step 1}} \frac{-4x^4+11x^3+9}{x^2} + \frac{4}{\color{red}{1}} = \frac{ -4x^4+11x^3+9 }{ x^2 } + \frac{ 4 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -4x^4+11x^3+9 } }{ x^2 } + \frac{ \color{purple}{ 4x^2 } }{ x^2 }=\frac{ \color{purple}{ -4x^4+11x^3+4x^2+9 } }{ x^2 } \end{aligned} $$