Multiply each term of $ \left( \color{blue}{x^2+4xy+4y^2}\right) $ by each term in $ \left( x-2y\right) $.

$$ \left( \color{blue}{x^2+4xy+4y^2}\right) \cdot \left( x-2y\right) = x^3-2x^2y+4x^2y-8xy^2+4xy^2-8y^3 $$

Combine like terms:

$$ x^3 \color{blue}{-2x^2y} + \color{blue}{4x^2y} \color{red}{-8xy^2} + \color{red}{4xy^2} -8y^3 = x^3+ \color{blue}{2x^2y} \color{red}{-4xy^2} -8y^3 $$

Multiply $a$ by $ \dfrac{x^3+2x^2y-4xy^2-8y^3}{x^3+2x^2y-4xy^2-8y^3} $ to get $ \dfrac{ ax^3+2ax^2y-4axy^2-8ay^3 }{ x^3+2x^2y-4xy^2-8y^3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} a \cdot \frac{x^3+2x^2y-4xy^2-8y^3}{x^3+2x^2y-4xy^2-8y^3} & \xlongequal{\text{Step 1}} \frac{a}{\color{red}{1}} \cdot \frac{x^3+2x^2y-4xy^2-8y^3}{x^3+2x^2y-4xy^2-8y^3} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ a \cdot \left( x^3+2x^2y-4xy^2-8y^3 \right) }{ 1 \cdot \left( x^3+2x^2y-4xy^2-8y^3 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ ax^3+2ax^2y-4axy^2-8ay^3 }{ x^3+2x^2y-4xy^2-8y^3 } \end{aligned} $$