$$\dfrac{a^3x^3+2a^3x^2y-4a^3xy^2-8a^3y^3}{a^2x^2+4a^2xy+4a^2y^2}\cdot\dfrac{1}{x-2y}=\dfrac{ax^3+2ax^2y-4axy^2-8ay^3}{x^3+2x^2y-4xy^2-8y^3}$$

Explanation

$$ \begin{aligned}\frac{a^3x^3+2a^3x^2y-4a^3xy^2-8a^3y^3}{a^2x^2+4a^2xy+4a^2y^2}\cdot\frac{1}{x-2y}& \xlongequal{ }\frac{ax^3+2ax^2y-4axy^2-8ay^3}{x^2+4xy+4y^2}\cdot\frac{1}{x-2y} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{ax^3+2ax^2y-4axy^2-8ay^3}{x^3+2x^2y-4xy^2-8y^3}\end{aligned} $$

①

Multiply $ \dfrac{ax^3+2ax^2y-4axy^2-8ay^3}{x^2+4xy+4y^2} $ by $ \dfrac{1}{x-2y} $ to get $ \dfrac{ax^3+2ax^2y-4axy^2-8ay^3}{x^3+2x^2y-4xy^2-8y^3} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

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

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