Divide :
`15a^(3)b^(4) - 10a^(4)b^(3) - 25a^(3)b^(6)` by `-5a^(3)b^(2)`

Answer

To divide the expression \( 15a^{3}b^{4} - 10a^{4}b^{3} - 25a^{3}b^{6} \) by \( -5a^{3}b^{2} \), we will follow these steps:

### Step 1: Write the expression
We need to divide the polynomial \( 15a^{3}b^{4} - 10a^{4}b^{3} - 25a^{3}b^{6} \) by \( -5a^{3}b^{2} \).

### Step 2: Separate the terms
We can separate the division into individual terms:
\[ \frac{15a^{3}b^{4}}{-5a^{3}b^{2}} - \frac{10a^{4}b^{3}}{-5a^{3}b^{2}} - \frac{25a^{3}b^{6}}{-5a^{3}b^{2}} \]

### Step 3: Simplify each term
Now, we will simplify each term separately.

1. **First Term**:
\[ \frac{15a^{3}b^{4}}{-5a^{3}b^{2}} = \frac{15}{-5} \cdot \frac{a^{3}}{a^{3}} \cdot \frac{b^{4}}{b^{2}} = -3 \cdot 1 \cdot b^{2} = -3b^{2} \]

2. **Second Term**:
\[ \frac{10a^{4}b^{3}}{-5a^{3}b^{2}} = \frac{10}{-5} \cdot \frac{a^{4}}{a^{3}} \cdot \frac{b^{3}}{b^{2}} = -2 \cdot a^{1} \cdot b^{1} = -2ab \]

3. **Third Term**:
\[ \frac{25a^{3}b^{6}}{-5a^{3}b^{2}} = \frac{25}{-5} \cdot \frac{a^{3}}{a^{3}} \cdot \frac{b^{6}}{b^{2}} = -5 \cdot 1 \cdot b^{4} = -5b^{4} \]

### Step 4: Combine the results
Now, we combine the simplified terms:
\[ -3b^{2} + 2ab + 5b^{4} \]

### Final Answer
Thus, the final result of the division is:
\[ 5b^{4} - 2ab - 3b^{2} \]