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Answer

$$\frac{1}{x^2}-4+\frac{6}{x}+2=\frac{-2x^3+6x^2+x}{x^3}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{1}{x^2}-4+\frac{6}{x}+2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-4x^2+1}{x^2}+\frac{6}{x}+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-4x^3+6x^2+x}{x^3}+2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-2x^3+6x^2+x}{x^3}\end{aligned} $$
①

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

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

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

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

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

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

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

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