The problem is to simplify the expression $\frac{-6x^4y^3 + 5x^2y^3 - 3x^3y}{-3x^2y}$.

1. Problem Description

The problem is to simplify the expression $\frac{-6x^4y^3 + 5x^2y^3 - 3x^3y}{-3x^2y}$.

2. Solution Steps

We divide each term in the numerator by the denominator.

$\frac{-6x^4y^3 + 5x^2y^3 - 3x^3y}{-3x^2y} = \frac{-6x^4y^3}{-3x^2y} + \frac{5x^2y^3}{-3x^2y} - \frac{3x^3y}{-3x^2y}$

Now simplify each term:

$\frac{-6x^4y^3}{-3x^2y} = \frac{-6}{-3} \cdot \frac{x^4}{x^2} \cdot \frac{y^3}{y} = 2x^{4-2}y^{3-1} = 2x^2y^2$

$\frac{5x^2y^3}{-3x^2y} = \frac{5}{-3} \cdot \frac{x^2}{x^2} \cdot \frac{y^3}{y} = -\frac{5}{3} x^{2-2} y^{3-1} = -\frac{5}{3} (1) y^2 = -\frac{5}{3} y^2$

$\frac{-3x^3y}{-3x^2y} = \frac{-3}{-3} \cdot \frac{x^3}{x^2} \cdot \frac{y}{y} = 1 \cdot x^{3-2} \cdot 1 = x$

So the expression becomes:

$2x^2y^2 - \frac{5}{3} y^2 + x$

3. Final Answer

$2x^2y^2 - \frac{5}{3}y^2 + x$