Fill in the blanks.
`(a+ 3b)^(2)` ______

Answer

To fill in the blank for the expression \((a + 3b)^{2}\), we will use the formula for the square of a binomial, which states that:

\[ (x + y)^{2} = x^{2} + y^{2} + 2xy \]

Here, we will let \(x = a\) and \(y = 3b\). Now, we can follow these steps to expand the expression:

### Step 1: Identify \(x\) and \(y\)
Let:
- \(x = a\)
- \(y = 3b\)

### Step 2: Apply the binomial square formula
Using the formula \((x + y)^{2} = x^{2} + y^{2} + 2xy\), we substitute \(x\) and \(y\):

\[ (a + 3b)^{2} = a^{2} + (3b)^{2} + 2(a)(3b) \]

### Step 3: Calculate each term
- The first term is \(a^{2}\).
- The second term is \((3b)^{2} = 9b^{2}\).
- The third term is \(2(a)(3b) = 6ab\).

### Step 4: Combine all terms
Now, we can combine all the terms:

\[ (a + 3b)^{2} = a^{2} + 9b^{2} + 6ab \]

### Final Answer
Thus, the expression \((a + 3b)^{2}\) can be filled in as:

\[ (a + 3b)^{2} = a^{2} + 9b^{2} + 6ab \]

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