Factor the following.
`4a^(2)+4ab+b^(2)=0`

Answer

To factor the quadratic expression \(4a^2 + 4ab + b^2\), we can follow these steps:

### Step 1: Identify the quadratic expression
We start with the expression:
\[ 4a^2 + 4ab + b^2 \]

### Step 2: Rewrite the middle term
We need to rewrite the middle term \(4ab\) in such a way that we can factor by grouping. We can split \(4ab\) into two equal parts:
\[ 4ab = 2ab + 2ab \]
Thus, we can rewrite the expression as:
\[ 4a^2 + 2ab + 2ab + b^2 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (4a^2 + 2ab) + (2ab + b^2) \]

### Step 4: Factor out the common terms from each group
From the first group \(4a^2 + 2ab\), we can factor out \(2a\):
\[ 2a(2a + b) \]
From the second group \(2ab + b^2\), we can factor out \(b\):
\[ b(2a + b) \]

### Step 5: Combine the factored terms
Now we can combine the factored expressions:
\[ 2a(2a + b) + b(2a + b) \]
We can see that \(2a + b\) is a common factor:
\[ (2a + b)(2a + b) \]

### Step 6: Write the final factored form
Thus, we can express the original quadratic as:
\[ (2a + b)^2 \]

### Conclusion
The factored form of the expression \(4a^2 + 4ab + b^2\) is:
\[ (2a + b)^2 = 0 \]