`16(x+y)^(2)-4y^(2)` =______

Options

A. `(4x-2y)(4x-2y)`
B. `(4x+2y)(4x-2y)`
C. `(4x-6y)(4x+2y)`
D. `(4x+2y)(4x+6y)`

Answer

To factorize the expression \( 16(x+y)^{2} - 4y^{2} \), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression can be recognized as a difference of squares. We can rewrite it in a more manageable form.

### Step 2: Rewrite the expression
We can express \( 16(x+y)^{2} - 4y^{2} \) as:
\[ (4(x+y))^{2} - (2y)^{2} \]

### Step 3: Apply the difference of squares formula
The difference of squares formula states that \( a^{2} - b^{2} = (a - b)(a + b) \). Here, let:
- \( a = 4(x+y) \)
- \( b = 2y \)

Using the formula, we get:
\[ (4(x+y) - 2y)(4(x+y) + 2y) \]

### Step 4: Simplify the factors
Now we simplify each factor:
1. For the first factor:
\[ 4(x+y) - 2y = 4x + 4y - 2y = 4x + 2y \]
2. For the second factor:
\[ 4(x+y) + 2y = 4x + 4y + 2y = 4x + 6y \]

### Step 5: Write the final factorized form
Putting it all together, we have:
\[ (4x + 2y)(4x + 6y) \]

Thus, the factorized form of \( 16(x+y)^{2} - 4y^{2} \) is:
\[ (4x + 2y)(4x + 6y) \]

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