`(x-y)^(2)+2xy=` ____

Options

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

Answer

To solve the expression \((x-y)^{2} + 2xy\), we can use the algebraic identity for the square of a binomial. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Binomial Square**:
We start with the expression \((x - y)^{2}\). According to the algebraic identity, we know that:
\[ (a - b)^{2} = a^{2} - 2ab + b^{2} \]
Here, \(a = x\) and \(b = y\).

2. **Expand the Binomial**:
Applying the identity:
\[ (x - y)^{2} = x^{2} - 2xy + y^{2} \]

3. **Substitute Back into the Expression**:
Now we substitute this expansion back into our original expression:
\[ (x - y)^{2} + 2xy = (x^{2} - 2xy + y^{2}) + 2xy \]

4. **Combine Like Terms**:
Now, we combine the like terms in the expression:
\[ x^{2} - 2xy + 2xy + y^{2} = x^{2} + y^{2} \]

5. **Final Result**:
Therefore, we find that:
\[ (x - y)^{2} + 2xy = x^{2} + y^{2} \]

### Conclusion:
The expression \((x - y)^{2} + 2xy\) simplifies to \(x^{2} + y^{2}\).