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

Options

A. `( a - 2 b - 4) ( a - 2 b + 2)`
B. ` ( a - b + 2) ( a- 4b - 4)`
C. ` (a + 2 b - 4) (a + 2 b + 2)`
D. ` (a + 2 b - 1) ( a - 2 b + 1)`

Answer

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

### Step 1: Rearrange the Expression
Rearranging the terms can sometimes make it easier to see how to factor. Let's write the expression in a more organized way:
\[ a^2 - 4ab - 2a + 4b^2 + 4b - 8 \]

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

### Step 3: Factor Each Group
Now, we will factor each group separately.

**For the first group \( a^2 - 4ab - 2a \):**
- We can factor out \( a \):
\[ a(a - 4b - 2) \]

**For the second group \( 4b^2 + 4b - 8 \):**
- We can factor out \( 4 \):
\[ 4(b^2 + b - 2) \]

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

### Step 5: Factor the Quadratic
Next, we need to factor \( b^2 + b - 2 \). We are looking for two numbers that multiply to \(-2\) and add to \(1\). The numbers \(2\) and \(-1\) work:
\[ b^2 + b - 2 = (b + 2)(b - 1) \]

### Step 6: Substitute Back
Now substituting back, we have:
\[ a(a - 4b - 2) + 4(b + 2)(b - 1) \]

### Step 7: Final Factorization
We can rewrite the entire expression as:
\[ (a - 2)(a - 4b + 4) \text{ or } (a - 2)(a + 4b - 4) \]

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