Which of the following is equal to `(x+i)^(2)-(x-i)^(2)`?

Options

A. `0`
B. `-2`
C. `-2+4x i`
D. `4x i`

Answer

To solve the expression \((x+i)^{2} - (x-i)^{2}\), we can use the difference of squares formula and expand both squares.

### Step-by-Step Solution:

1. **Identify the expression**: We have \((x+i)^{2} - (x-i)^{2}\).

2. **Use the formula for the square of a binomial**:
\[ (a+b)^{2} = a^{2} + 2ab + b^{2} \]
\[ (a-b)^{2} = a^{2} - 2ab + b^{2} \]

Here, let \(a = x\) and \(b = i\).

3. **Expand \((x+i)^{2}\)**:
\[ (x+i)^{2} = x^{2} + 2xi + i^{2} \]
Since \(i^{2} = -1\), we can write:
\[ (x+i)^{2} = x^{2} + 2xi - 1 \]

4. **Expand \((x-i)^{2}\)**:
\[ (x-i)^{2} = x^{2} - 2xi + i^{2} \]
Again, substituting \(i^{2} = -1\):
\[ (x-i)^{2} = x^{2} - 2xi - 1 \]

5. **Subtract the two expansions**:
\[ (x+i)^{2} - (x-i)^{2} = (x^{2} + 2xi - 1) - (x^{2} - 2xi - 1) \]

6. **Simplify the expression**:
\[ = x^{2} + 2xi - 1 - x^{2} + 2xi + 1 \]
The \(x^{2}\) terms cancel out and the \(-1\) and \(+1\) also cancel:
\[ = 2xi + 2xi = 4xi \]

7. **Final result**:
\[ (x+i)^{2} - (x-i)^{2} = 4xi \]

### Final Answer:
\[ 4xi \]