The problem is to divide the polynomial $ -\frac{3}{4}x^{n-1}y^{n+2} + \frac{1}{8}x^{n+1} - \frac{2}{3}x^{n+1}y^m $ by the monomial $ -\frac{2}{5}x^3y^2 $.

1. Problem Description

The problem is to divide the polynomial $-\frac{3}{4}x^{n-1}y^{n+2} + \frac{1}{8}x^{n+1} - \frac{2}{3}x^{n+1}y^m$ by the monomial $-\frac{2}{5}x^3y^2$.

2. Solution Steps

First, we write the division:

$\frac{-\frac{3}{4}x^{n-1}y^{n+2} + \frac{1}{8}x^{n+1} - \frac{2}{3}x^{n+1}y^m}{-\frac{2}{5}x^3y^2}$

Next, we divide each term of the polynomial by the monomial.

$\frac{-\frac{3}{4}x^{n-1}y^{n+2}}{-\frac{2}{5}x^3y^2} = \frac{-\frac{3}{4}}{-\frac{2}{5}} \cdot \frac{x^{n-1}}{x^3} \cdot \frac{y^{n+2}}{y^2} = \frac{3}{4} \cdot \frac{5}{2} \cdot x^{n-1-3} \cdot y^{n+2-2} = \frac{15}{8}x^{n-4}y^n$

$\frac{\frac{1}{8}x^{n+1}}{-\frac{2}{5}x^3y^2} = \frac{\frac{1}{8}}{-\frac{2}{5}} \cdot \frac{x^{n+1}}{x^3} \cdot \frac{1}{y^2} = \frac{1}{8} \cdot (-\frac{5}{2}) \cdot x^{n+1-3} \cdot y^{-2} = -\frac{5}{16}x^{n-2}y^{-2}$

$\frac{-\frac{2}{3}x^{n+1}y^m}{-\frac{2}{5}x^3y^2} = \frac{-\frac{2}{3}}{-\frac{2}{5}} \cdot \frac{x^{n+1}}{x^3} \cdot \frac{y^m}{y^2} = \frac{2}{3} \cdot \frac{5}{2} \cdot x^{n+1-3} \cdot y^{m-2} = \frac{5}{3}x^{n-2}y^{m-2}$

Therefore, the result of the division is:

$\frac{15}{8}x^{n-4}y^n -\frac{5}{16}x^{n-2}y^{-2} + \frac{5}{3}x^{n-2}y^{m-2}$

3. Final Answer

$\frac{15}{8}x^{n-4}y^n -\frac{5}{16}x^{n-2}y^{-2} + \frac{5}{3}x^{n-2}y^{m-2}$