The factors `8(a-2b)^(2)-2a+4b-1=` are

Options

A. `(2a-4b-1)(4a-8b+1)`
B. `(2a+4b+1)(4a-8b+1)`
C. `(2a-4b-1)(4a-8b-1)`
D. `(2a+4b-1)(4a+8b+1)`

Answer

To factor the expression \( 8(a - 2b)^2 - 2a + 4b - 1 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 8(a - 2b)^2 - 2a + 4b - 1 \]

### Step 2: Substitute a variable
Let \( x = a - 2b \). Then, we can rewrite the expression as:
\[ 8x^2 - 2a + 4b - 1 \]

### Step 3: Express \( a \) and \( b \) in terms of \( x \)
From our substitution, we know:
\[ a = x + 2b \]
Substituting this into the expression gives:
\[ 8x^2 - 2(x + 2b) + 4b - 1 \]
This simplifies to:
\[ 8x^2 - 2x - 4b + 4b - 1 = 8x^2 - 2x - 1 \]

### Step 4: Factor the quadratic expression
Now we need to factor \( 8x^2 - 2x - 1 \). To do this, we look for two numbers that multiply to \( 8 \times -1 = -8 \) and add to \( -2 \). The numbers that work are \( -4 \) and \( 2 \).

### Step 5: Rewrite the middle term
We can rewrite the expression as:
\[ 8x^2 - 4x + 2x - 1 \]

### Step 6: Group the terms
Now we group the terms:
\[ (8x^2 - 4x) + (2x - 1) \]

### Step 7: Factor by grouping
Factoring out common terms from each group gives us:
\[ 4x(2x - 1) + 1(2x - 1) \]
Now we can factor out \( (2x - 1) \):
\[ (2x - 1)(4x + 1) \]

### Step 8: Substitute back for \( x \)
Now we substitute back \( x = a - 2b \):
\[ (2(a - 2b) - 1)(4(a - 2b) + 1) \]
This simplifies to:
\[ (2a - 4b - 1)(4a - 8b + 1) \]

### Final Answer
Thus, the factors of the expression \( 8(a - 2b)^2 - 2a + 4b - 1 \) are:
\[ (2a - 4b - 1)(4a - 8b + 1) \]