Multiply $ \color{blue}{x^2} $ by $ \left( a-x-y\right) $ $$ \color{blue}{x^2} \cdot \left( a-x-y\right) = ax^2-x^3-x^2y $$Multiply $ \color{blue}{y^2} $ by $ \left( a-x-y\right) $ $$ \color{blue}{y^2} \cdot \left( a-x-y\right) = ay^2-xy^2-y^3 $$

Multiply each term of $ \left( \color{blue}{a-x-y}\right) $ by each term in $ \left( a-x-y\right) $.

$$ \left( \color{blue}{a-x-y}\right) \cdot \left( a-x-y\right) = a^2-ax-ay-ax+x^2+xy-ay+xy+y^2 $$

Combine like terms:

$$ a^2 \color{blue}{-ax} \color{red}{-ay} \color{blue}{-ax} +x^2+ \color{green}{xy} \color{red}{-ay} + \color{green}{xy} +y^2 = \\ = a^2 \color{blue}{-2ax} \color{red}{-2ay} +x^2+ \color{green}{2xy} +y^2 $$

Multiply each term of $ \left( \color{blue}{a^2-2ax-2ay+x^2+2xy+y^2}\right) $ by each term in $ \left( a-x-y\right) $.

$$ \left( \color{blue}{a^2-2ax-2ay+x^2+2xy+y^2}\right) \cdot \left( a-x-y\right) = \\ = a^3-a^2x-a^2y-2a^2x+2ax^2+2axy-2a^2y+2axy+2ay^2+ax^2-x^3-x^2y+2axy-2x^2y-2xy^2+ay^2-xy^2-y^3 $$

Combine like terms:

$$ a^3 \color{blue}{-a^2x} \color{red}{-a^2y} \color{blue}{-2a^2x} + \color{green}{2ax^2} + \color{orange}{2axy} \color{red}{-2a^2y} + \color{blue}{2axy} + \color{red}{2ay^2} + \color{green}{ax^2} -x^3 \color{green}{-x^2y} + \color{blue}{2axy} \color{green}{-2x^2y} \color{orange}{-2xy^2} + \color{red}{ay^2} \color{orange}{-xy^2} -y^3 = \\ = a^3 \color{blue}{-3a^2x} \color{red}{-3a^2y} + \color{green}{3ax^2} + \color{blue}{6axy} + \color{red}{3ay^2} -x^3 \color{green}{-3x^2y} \color{orange}{-3xy^2} -y^3 $$

Add $ \dfrac{a^3-3a^2x-3a^2y+3ax^2+6axy+3ay^2-x^3-3x^2y-3xy^2-y^3}{3} $ and $ ax^2-x^3-x^2y $ to get $ \dfrac{ \color{purple}{ a^3-3a^2x-3a^2y+6ax^2+6axy+3ay^2-4x^3-6x^2y-3xy^2-y^3 } }{ 3 }$.

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

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

Add $ \dfrac{a^3-3a^2x-3a^2y+6ax^2+6axy+3ay^2-4x^3-6x^2y-3xy^2-y^3}{3} $ and $ ay^2-xy^2-y^3 $ to get $ \dfrac{ \color{purple}{ a^3-3a^2x-3a^2y+6ax^2+6axy+6ay^2-4x^3-6x^2y-6xy^2-4y^3 } }{ 3 }$.

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

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