The factors `(a ^(2) + 4b ^(2) + 4b - 4 ab - 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. `(a + 2b - 1) (a - 2b + 1)`

Answer

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

### Step 1: Rearrange the Expression
First, we rewrite the expression in a more manageable form:
\[ a^2 - 4ab + 4b^2 - 2a + 4b - 8 \]

### Step 2: Group Terms
Next, we can group the terms to facilitate factoring:
\[ (a^2 - 4ab + 4b^2) + (-2a + 4b - 8) \]

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

### Step 4: Factor the Second Group
Now, we look at the second group \( -2a + 4b - 8 \). We can factor out \(-2\):
\[ -2(a - 2b + 4) \]

### Step 5: Combine the Factors
Now we can combine our factored forms:
\[ (a - 2b)^2 - 2(a - 2b + 4) \]

### Step 6: Substitute \( t = a - 2b \)
Let \( t = a - 2b \). Then we can rewrite the expression as:
\[ t^2 - 2(t + 4) \]

### Step 7: Simplify
This simplifies to:
\[ t^2 - 2t - 8 \]

### Step 8: Factor the Quadratic
Now we can factor the quadratic \( t^2 - 2t - 8 \):
\[ (t - 4)(t + 2) \]

### Step 9: Substitute Back
Substituting back \( t = a - 2b \):
\[ (a - 2b - 4)(a - 2b + 2) \]

### Final Factored Form
Thus, the complete factored form of the original expression is:
\[ (a - 2b - 4)(a - 2b + 2) \]

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