The problem asks us to divide the polynomial $27x^2y - 9xy^2 + 6x^2y^2$ by $-3xy$.

1. Problem Description

The problem asks us to divide the polynomial $27x^2y - 9xy^2 + 6x^2y^2$ by $-3xy$.

2. Solution Steps

We need to divide each term of the polynomial by $-3xy$.

$\frac{27x^2y - 9xy^2 + 6x^2y^2}{-3xy} = \frac{27x^2y}{-3xy} - \frac{9xy^2}{3xy} + \frac{6x^2y^2}{-3xy}$

Now, we simplify each fraction:

$\frac{27x^2y}{-3xy} = \frac{27}{-3} \cdot \frac{x^2}{x} \cdot \frac{y}{y} = -9x$

$\frac{-9xy^2}{-3xy} = \frac{-9}{-3} \cdot \frac{x}{x} \cdot \frac{y^2}{y} = 3y$

$\frac{6x^2y^2}{-3xy} = \frac{6}{-3} \cdot \frac{x^2}{x} \cdot \frac{y^2}{y} = -2xy$

So the result is:

$-9x + 3y - 2xy$

3. Final Answer

The final answer is $-9x + 3y - 2xy$.