Find the product of `3ab^(2)and (4ab+3b^(2))`

Answer

To find the product of \(3ab^2\) and \((4ab + 3b^2)\), we will use the distributive property, which states that \(a(b + c) = ab + ac\).

### Step-by-step Solution:

1. **Identify the Expression**:
We need to find the product of \(3ab^2\) and \((4ab + 3b^2)\).

2. **Apply the Distributive Property**:
We can distribute \(3ab^2\) to both terms in the parentheses:
\[ 3ab^2 \cdot (4ab + 3b^2) = 3ab^2 \cdot 4ab + 3ab^2 \cdot 3b^2 \]

3. **Multiply the First Term**:
Now, we will calculate \(3ab^2 \cdot 4ab\):
- Multiply the coefficients: \(3 \cdot 4 = 12\)
- Multiply the \(a\) terms: \(a \cdot a = a^{1+1} = a^2\)
- Multiply the \(b\) terms: \(b^2 \cdot b = b^{2+1} = b^3\)
- So, \(3ab^2 \cdot 4ab = 12a^2b^3\)

4. **Multiply the Second Term**:
Next, we will calculate \(3ab^2 \cdot 3b^2\):
- Multiply the coefficients: \(3 \cdot 3 = 9\)
- The \(a\) term remains the same: \(a\)
- Multiply the \(b\) terms: \(b^2 \cdot b^2 = b^{2+2} = b^4\)
- So, \(3ab^2 \cdot 3b^2 = 9ab^4\)

5. **Combine the Results**:
Now, we combine both results:
\[ 12a^2b^3 + 9ab^4 \]

### Final Answer:
The product of \(3ab^2\) and \((4ab + 3b^2)\) is:
\[ 12a^2b^3 + 9ab^4 \]