Subtract $ \dfrac{18}{y^2} $ from $ y^2+7y $ to get $ \dfrac{ \color{purple}{ y^4+7y^3-18 } }{ y^2 }$.

Step 1: Write $ y^2+7y $ 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 $\color{blue}{ y^2 }$.

$$ \begin{aligned} y^2+7y- \frac{18}{y^2} & \xlongequal{\text{Step 1}} \frac{y^2+7y}{\color{red}{1}} - \frac{18}{y^2} = \frac{ \left( y^2+7y \right) \cdot \color{blue}{ y^2 }}{ 1 \cdot \color{blue}{ y^2 }} - \frac{ 18 }{ y^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ y^4+7y^3 } }{ y^2 } - \frac{ \color{purple}{ 18 } }{ y^2 }=\frac{ \color{purple}{ y^4+7y^3-18 } }{ y^2 } \end{aligned} $$

Subtract $3y$ from $ \dfrac{y^4+7y^3-18}{y^2} $ to get $ \dfrac{ \color{purple}{ y^4+4y^3-18 } }{ y^2 }$.

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

$$ \begin{aligned} \frac{y^4+7y^3-18}{y^2} -3y & \xlongequal{\text{Step 1}} \frac{y^4+7y^3-18}{y^2} - \frac{3y}{\color{red}{1}} = \frac{ y^4+7y^3-18 }{ y^2 } - \frac{ 3y \cdot \color{blue}{ y^2 }}{ 1 \cdot \color{blue}{ y^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ y^4+7y^3-18 } }{ y^2 } - \frac{ \color{purple}{ 3y^3 } }{ y^2 }=\frac{ \color{purple}{ y^4+4y^3-18 } }{ y^2 } \end{aligned} $$

Add $ \dfrac{y^4+4y^3-18}{y^2} $ and $ 2 $ to get $ \dfrac{ \color{purple}{ y^4+4y^3+2y^2-18 } }{ y^2 }$.

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 $\color{blue}{y^2}$.

$$ \begin{aligned} \frac{y^4+4y^3-18}{y^2} +2 & \xlongequal{\text{Step 1}} \frac{y^4+4y^3-18}{y^2} + \frac{2}{\color{red}{1}} = \frac{ y^4+4y^3-18 }{ y^2 } + \frac{ 2 \cdot \color{blue}{ y^2 }}{ 1 \cdot \color{blue}{ y^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ y^4+4y^3-18 } }{ y^2 } + \frac{ \color{purple}{ 2y^2 } }{ y^2 }=\frac{ \color{purple}{ y^4+4y^3+2y^2-18 } }{ y^2 } \end{aligned} $$