Which of the following correct ?

Options

A. `(5x+y) ^(2) = 25 x ^(2) + 5xy + y ^(2)`
B. `(5x -y) ^(2) = 25 x ^(2) - 10 xy + y ^(2)`
C. `(5x - y) ^(2) = 25 x ^(2) + 10xy + y ^(2)`
D. `(5x + y) ^(2) = 25 x ^(2) + 5xy + y ^(2)`

Answer

To determine which of the given options is correct, we need to analyze the expressions involving \( A \) and \( B \), where \( A = 5x \) and \( B = y \). We will evaluate \( (5x + y)^2 \) and \( (5x - y)^2 \) using the algebraic identity for the square of a binomial.

### Step-by-Step Solution:

1. **Identify the expressions:**
- We need to evaluate \( (5x + y)^2 \) and \( (5x - y)^2 \).

2. **Apply the binomial square formula:**
- The formula for the square of a binomial is:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

3. **Calculate \( (5x + y)^2 \):**
- Here, \( a = 5x \) and \( b = y \).
- Applying the formula:
\[ (5x + y)^2 = (5x)^2 + 2(5x)(y) + y^2 = 25x^2 + 10xy + y^2 \]

4. **Calculate \( (5x - y)^2 \):**
- Again, using \( a = 5x \) and \( b = y \):
\[ (5x - y)^2 = (5x)^2 - 2(5x)(y) + y^2 = 25x^2 - 10xy + y^2 \]

5. **Compare the results with the options:**
- From the calculations, we have:
- \( (5x + y)^2 = 25x^2 + 10xy + y^2 \)
- \( (5x - y)^2 = 25x^2 - 10xy + y^2 \)
- We need to check the options provided:
- **Option A:** \( 5x + y \) (not a square)
- **Option B:** \( (5x - y)^2 = 25x^2 - 10xy + y^2 \) (correct)
- **Option C:** \( (5x + y)^2 = 25x^2 + 10xy + y^2 \) (not a match)
- **Option D:** Incorrect format (missing terms)

6. **Conclusion:**
- The correct option is **B**, which corresponds to \( (5x - y)^2 \).