Add $x^2+7x$ and $ \dfrac{6}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4+7x^3+6 } }{ x^2 }$.

Step 1: Write $ x^2+7x $ 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 first fraction by $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^2+7x+ \frac{6}{x^2} & \xlongequal{\text{Step 1}} \frac{x^2+7x}{\color{red}{1}} + \frac{6}{x^2} = \frac{ \left( x^2+7x \right) \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 6 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+7x^3 } }{ x^2 } + \frac{ \color{purple}{ 6 } }{ x^2 }=\frac{ \color{purple}{ x^4+7x^3+6 } }{ x^2 } \end{aligned} $$

Subtract $3x$ from $ \dfrac{x^4+7x^3+6}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4+4x^3+6 } }{ x^2 }$.

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

$$ \begin{aligned} \frac{x^4+7x^3+6}{x^2} -3x & \xlongequal{\text{Step 1}} \frac{x^4+7x^3+6}{x^2} - \frac{3x}{\color{red}{1}} = \frac{ x^4+7x^3+6 }{ x^2 } - \frac{ 3x \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4+7x^3+6 } }{ x^2 } - \frac{ \color{purple}{ 3x^3 } }{ x^2 }=\frac{ \color{purple}{ x^4+4x^3+6 } }{ x^2 } \end{aligned} $$

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

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