Subtract $x$ from $ \dfrac{x^2+5x+6}{x^2} $ to get $ \dfrac{ \color{purple}{ -x^3+x^2+5x+6 } }{ x^2 }$.

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

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

Subtract $6$ from $ \dfrac{-x^3+x^2+5x+6}{x^2} $ to get $ \dfrac{ \color{purple}{ -x^3-5x^2+5x+6 } }{ x^2 }$.

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

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