The factors of `a ^(2) -b ^(2) - 4c ^(2) + 4d ^(2) - 4 (ad - bc) ` are :

Options

A. `(a + 2d + b + 2c) (a - 2d - b + 2c)`
B. `( a- 2d + b - 2c) (a + 2d - b + 2c)`
C. `(a - 2d + b - 2c) (a - 2d - b + 2c)`
D. `(a - 2d - b- 2c) (a + 2d+ b + 2c)`

Answer

To factor the expression \( a^2 - b^2 - 4c^2 + 4d^2 - 4(ad - bc) \), we can follow these steps:

### Step 1: Rewrite the expression
Start with the given expression:
\[ a^2 - b^2 - 4c^2 + 4d^2 - 4(ad - bc) \]
Distributing the \(-4\) inside the parentheses gives:
\[ a^2 - b^2 - 4c^2 + 4d^2 - 4ad + 4bc \]

### Step 2: Rearrange the terms
Now, rearranging the expression:
\[ a^2 - 4ad + 4d^2 - b^2 + 4bc - 4c^2 \]

### Step 3: Group the terms
Group the first three terms and the last three terms:
\[ (a^2 - 4ad + 4d^2) + (-b^2 + 4bc - 4c^2) \]

### Step 4: Factor the perfect squares
Recognize that \(a^2 - 4ad + 4d^2\) is a perfect square:
\[ (a - 2d)^2 \]
And \(-b^2 + 4bc - 4c^2\) can be rewritten as:
\[ -(b^2 - 4bc + 4c^2) = -(b - 2c)^2 \]
Thus, we have:
\[ (a - 2d)^2 - (b - 2c)^2 \]

### Step 5: Apply the difference of squares formula
Now, we can apply the difference of squares formula \(x^2 - y^2 = (x + y)(x - y)\):
Let \(x = (a - 2d)\) and \(y = (b - 2c)\):
\[ (a - 2d + b - 2c)(a - 2d - (b - 2c)) \]

### Step 6: Simplify the factors
Now simplify the factors:
1. \( (a - 2d + b - 2c) = (a + b - 2d - 2c) \)
2. \( (a - 2d - (b - 2c)) = (a - b + 2c - 2d) \)

Thus, the final factored form is:
\[ (a + b - 2d - 2c)(a - b + 2c - 2d) \]

### Final Answer
The factors of the expression \( a^2 - b^2 - 4c^2 + 4d^2 - 4(ad - bc) \) are:
\[ (a + b - 2d - 2c)(a - b + 2c - 2d) \]