`4a^2+b^2-4ab+8a-4b+4= ? `

Options

A. `(2a+b+2)^2`
B. `(2a-b+2)^2`
C. `(a+2b+2)^2`
D. None of these

Explanation

To factor the expression \( 4a^2 + b^2 - 4ab + 8a - 4b + 4 \), we will follow these steps:

### Step 1: Rearrange the Expression
First, we can rearrange the terms of the expression for clarity:
\[ 4a^2 - 4ab + b^2 + 8a - 4b + 4 \]

### Step 2: Group the Terms
Next, we can group the terms that can be factored together:
\[ (4a^2 - 4ab + b^2) + (8a - 4b + 4) \]

### Step 3: Factor the First Group
Now, we will focus on the first group \( 4a^2 - 4ab + b^2 \). This can be recognized as a perfect square:
\[ 4a^2 - 4ab + b^2 = (2a - b)^2 \]

### Step 4: Factor the Second Group
Next, we will factor the second group \( 8a - 4b + 4 \):
\[ 8a - 4b + 4 = 4(2a - b + 1) \]

### Step 5: Combine the Factors
Now we can combine the factored forms:
\[ (2a - b)^2 + 4(2a - b + 1) \]
Notice that \( 2a - b \) is a common term. Let’s denote \( x = 2a - b \):
\[ x^2 + 4(x + 1) \]

### Step 6: Simplify
Now, we can simplify this expression:
\[ x^2 + 4x + 4 = (x + 2)^2 \]

### Step 7: Substitute Back
Substituting back \( x = 2a - b \):
\[ (2a - b + 2)^2 \]

### Final Answer
Thus, the expression \( 4a^2 + b^2 - 4ab + 8a - 4b + 4 \) can be factored as:
\[ (2a - b + 2)^2 \]