Multiply $9$ by $ \dfrac{x^2}{y^2} $ to get $ \dfrac{ 9x^2 }{ y^2 } $.

Step 1: Write $ 9 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 9 \cdot \frac{x^2}{y^2} & \xlongequal{\text{Step 1}} \frac{9}{\color{red}{1}} \cdot \frac{x^2}{y^2} \xlongequal{\text{Step 2}} \frac{ 9 \cdot x^2 }{ 1 \cdot y^2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 9x^2 }{ y^2 } \end{aligned} $$

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

Step 1: Write $ y^2+6xy $ 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 $\color{blue}{ y^2 }$.

$$ \begin{aligned} y^2+6xy+ \frac{9x^2}{y^2} & \xlongequal{\text{Step 1}} \frac{y^2+6xy}{\color{red}{1}} + \frac{9x^2}{y^2} = \frac{ \left( y^2+6xy \right) \cdot \color{blue}{ y^2 }}{ 1 \cdot \color{blue}{ y^2 }} + \frac{ 9x^2 }{ y^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ y^4+6xy^3 } }{ y^2 } + \frac{ \color{purple}{ 9x^2 } }{ y^2 }=\frac{ \color{purple}{ 6xy^3+y^4+9x^2 } }{ y^2 } \end{aligned} $$

Subtract $9x^2$ from $ \dfrac{6xy^3+y^4+9x^2}{y^2} $ to get $ \dfrac{ \color{purple}{ -9x^2y^2+6xy^3+y^4+9x^2 } }{ y^2 }$.

Step 1: Write $ 9x^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 $\color{blue}{y^2}$.

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