The value of ` (2 a + b)^(2) - (2 a - b)^( 2) ` is

Options

A. 8ab
B. `-8ab`
C. `8a^(2) + 2b^(2)`
D. `8a^(2) - 2b^(2)`

Answer

To solve the expression \( (2a + b)^2 - (2a - b)^2 \), we can use the difference of squares formula.

### Step 1: Identify the formula
The difference of squares states that:
\[ x^2 - y^2 = (x - y)(x + y) \]
In our case, let \( x = (2a + b) \) and \( y = (2a - b) \).

### Step 2: Apply the formula
Using the difference of squares, we can rewrite the expression:
\[ (2a + b)^2 - (2a - b)^2 = [(2a + b) - (2a - b)][(2a + b) + (2a - b)] \]

### Step 3: Simplify the first factor
Now, simplify the first factor:
\[ (2a + b) - (2a - b) = 2a + b - 2a + b = 2b \]

### Step 4: Simplify the second factor
Next, simplify the second factor:
\[ (2a + b) + (2a - b) = 2a + b + 2a - b = 4a \]

### Step 5: Combine the factors
Now, we can combine the results:
\[ (2b)(4a) = 8ab \]

### Conclusion
Thus, the value of \( (2a + b)^2 - (2a - b)^2 \) is:
\[ \boxed{8ab} \]