`(a+b)^(2)-(b-a)^(2)`=_______

Options

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

Answer

To solve the expression \((a+b)^{2} - (b-a)^{2}\), we can use the difference of squares identity, which states that \(A^{2} - B^{2} = (A + B)(A - B)\).

### Step-by-Step Solution:

1. **Identify A and B:**
Let \(A = (a+b)\) and \(B = (b-a)\).

2. **Apply the Difference of Squares Identity:**
Using the identity \(A^{2} - B^{2} = (A + B)(A - B)\), we can rewrite our expression:
\[ (a+b)^{2} - (b-a)^{2} = [(a+b) + (b-a)][(a+b) - (b-a)] \]

3. **Simplify the First Part \((A + B)\):**
\[ (a+b) + (b-a) = a + b + b - a = 2b \]

4. **Simplify the Second Part \((A - B)\):**
\[ (a+b) - (b-a) = a + b - b + a = 2a \]

5. **Combine the Results:**
Now we can substitute back into our expression:
\[ (a+b)^{2} - (b-a)^{2} = (2b)(2a) = 4ab \]

### Final Answer:
Thus, \((a+b)^{2} - (b-a)^{2} = 4ab\).

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