If equation `4x^(2)+2pxy+25y^(2)+2x+5y-1=0`
represents parallel lines, then p is equal to

Options

A. `-10`
B. `10`
C. 5
D. `-2`

Explanation

To solve the problem, we need to determine the value of \( p \) such that the equation

\[ 4x^2 + 2pxy + 25y^2 + 2x + 5y - 1 = 0 \]

represents parallel lines.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation can be compared to the general form of a conic section:

\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

From the equation, we identify:
- \( a = 4 \)
- \( h = p \)
- \( b = 25 \)
- \( g = 1 \)
- \( f = \frac{5}{2} \)
- \( c = -1 \)

2. **Condition for parallel lines**:
For the equation to represent parallel lines, the condition is:

\[ h^2 = ab \]

Substituting the values we identified:
\[ p^2 = 4 \cdot 25 \]

3. **Calculate \( ab \)**:
Calculate \( 4 \cdot 25 \):
\[ ab = 100 \]

4. **Set up the equation**:
Now we have:
\[ p^2 = 100 \]

5. **Solve for \( p \)**:
Taking the square root of both sides gives:
\[ p = \pm 10 \]

### Conclusion:
Thus, the values of \( p \) that make the equation represent parallel lines are \( p = 10 \) or \( p = -10 \).