Divide (20x8y3 − 12x5y2) ÷ (−4x2y). −5x6y3 + 3x3y2 5x6y3 − 3x3y2 −5x6y2 + 3x3y 5x6y2 − 3x3y

1. Break Down the Problem

We need to divide the expression $20x^8y^3 - 12x^5y^2$ by $-4x^2y$. This involves separating the division operation and simplifying each term in the numerator.

2. Relevant Concepts

The division of polynomials can be done by dividing each term of the polynomial in the numerator by the polynomial in the denominator. The terms will be simplified individually.

3. Analysis and Detail

Let's divide each term of the numerator by the denominator:

1. First Term: $\frac{20x^8y^3}{-4x^2y} = \frac{20}{-4} \cdot \frac{x^8}{x^2} \cdot \frac{y^3}{y} = -5 \cdot x^{8-2} \cdot y^{3-1} = -5x^6y^2$

2. Second Term: $\frac{-12x^5y^2}{-4x^2y} = \frac{-12}{-4} \cdot \frac{x^5}{x^2} \cdot \frac{y^2}{y} = 3 \cdot x^{5-2} \cdot y^{2-1} = 3x^3y$

Now, we combine the simplified terms:

$-5x^6y^2 + 3x^3y$

4. Verify and Summarize

Both terms have been simplified correctly. The division has been completed, and the final expression is:

Final Answer

$-5x^6y^2 + 3x^3y$