Multiply $y^2$ by $ \dfrac{1-x^4y}{x^2} $ to get $ \dfrac{-x^4y^3+y^2}{x^2} $.

Step 1: Write $ y^2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ 9x^4y^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 first fraction by $\color{blue}{ x^2 }$.

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

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

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

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