The factors of `(a^2+4b^2+4b-4ab-2a-8)` are

Options

A. (a-2b-4)(a-2b+2)
B. (a-b+2)(a-4b-4)
C. (a+2b-4)(a+2b+2)
D. none of these

Answer

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

### Step 1: Rearrange the Expression
Start by rearranging the terms in the expression for clarity:
\[ a^2 - 4ab + 4b^2 + 4b - 2a - 8 \]

### Step 2: Group the Terms
Next, we can group the terms in a way that helps us to factor:
\[ (a^2 - 4ab + 4b^2) + (4b - 2a - 8) \]

### Step 3: Factor the First Group
The first group \( a^2 - 4ab + 4b^2 \) can be recognized as a perfect square:
\[ a^2 - 4ab + 4b^2 = (a - 2b)^2 \]

### Step 4: Simplify the Second Group
Now, let's simplify the second group \( 4b - 2a - 8 \):
\[ 4b - 2a - 8 = -2(a - 2b + 4) \]

### Step 5: Combine the Factors
Now we can rewrite the expression using the factors we found:
\[ (a - 2b)^2 - 2(a - 2b + 4) \]

### Step 6: Recognize the Structure
Notice that we can express this in a form that resembles a difference of squares:
\[ (a - 2b - 1)^2 - 3^2 \]

### Step 7: Apply the Difference of Squares Formula
Using the difference of squares formula \( x^2 - y^2 = (x - y)(x + y) \):
Let \( x = (a - 2b - 1) \) and \( y = 3 \):
\[ = (a - 2b - 1 - 3)(a - 2b - 1 + 3) \]
\[ = (a - 2b - 4)(a - 2b + 2) \]

### Final Answer
Thus, the factors of the expression \( a^2 + 4b^2 + 4b - 4ab - 2a - 8 \) are:
\[ (a - 2b - 4)(a - 2b + 2) \]

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