Divide `14x^(3)y^(2)+8x^(2)y^(3)-32x^(2)y^(5)` by `-2xy^(2)`

Answer

To divide the expression \( 14x^3y^2 + 8x^2y^3 - 32x^2y^5 \) by \( -2xy^2 \), we will perform the division term by term.

### Step-by-Step Solution:

1. **Write down the expression to be divided**:
\[ \frac{14x^3y^2 + 8x^2y^3 - 32x^2y^5}{-2xy^2} \]

2. **Divide the first term**:
- Divide \( 14x^3y^2 \) by \( -2xy^2 \):
\[ \frac{14x^3y^2}{-2xy^2} = \frac{14}{-2} \cdot \frac{x^3}{x} \cdot \frac{y^2}{y^2} = -7x^{3-1} = -7x^2 \]

3. **Divide the second term**:
- Divide \( 8x^2y^3 \) by \( -2xy^2 \):
\[ \frac{8x^2y^3}{-2xy^2} = \frac{8}{-2} \cdot \frac{x^2}{x} \cdot \frac{y^3}{y^2} = -4x^{2-1}y^{3-2} = -4xy \]

4. **Divide the third term**:
- Divide \( -32x^2y^5 \) by \( -2xy^2 \):
\[ \frac{-32x^2y^5}{-2xy^2} = \frac{-32}{-2} \cdot \frac{x^2}{x} \cdot \frac{y^5}{y^2} = 16x^{2-1}y^{5-2} = 16xy^3 \]

5. **Combine the results**:
- After dividing all the terms, we combine the results:
\[ -7x^2 - 4xy + 16xy^3 \]

### Final Answer:
The result of dividing \( 14x^3y^2 + 8x^2y^3 - 32x^2y^5 \) by \( -2xy^2 \) is:
\[ -7x^2 - 4xy + 16xy^3 \]