Divide `14x^(3)y^(2) + 8x^(2)y^(3) - 22xy^(4) " by " -2xy^(2)`

Answer

To solve the problem of dividing the expression \(14x^3y^2 + 8x^2y^3 - 22xy^4\) by \(-2xy^2\), we will follow these steps:

### Step 1: Write the expression to be divided
We start with the expression:
\[ \frac{14x^3y^2 + 8x^2y^3 - 22xy^4}{-2xy^2} \]

### Step 2: Split the division
We can split the division into separate fractions:
\[ = \frac{14x^3y^2}{-2xy^2} + \frac{8x^2y^3}{-2xy^2} - \frac{22xy^4}{-2xy^2} \]

### Step 3: Simplify each term
Now we simplify each term individually.

1. **First term:**
\[ \frac{14x^3y^2}{-2xy^2} = 14 \div -2 \cdot \frac{x^3}{x} \cdot \frac{y^2}{y^2} = -7x^{3-1} = -7x^2 \]

2. **Second term:**
\[ \frac{8x^2y^3}{-2xy^2} = 8 \div -2 \cdot \frac{x^2}{x} \cdot \frac{y^3}{y^2} = -4x^{2-1}y^{3-2} = -4xy \]

3. **Third term:**
\[ -\frac{22xy^4}{-2xy^2} = 22 \div -2 \cdot \frac{xy^4}{xy^2} = 11y^{4-2} = 11y^2 \]

### Step 4: Combine the results
Now we combine the simplified terms:
\[ -7x^2 - 4xy + 11y^2 \]

### Final Result
Thus, the final result of the division is:
\[ -7x^2 - 4xy + 11y^2 \]

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