One of the factors of `(25 x^(2)-1) +(1+5x)^(2)` is

Options

A. `5+x`
B. `5-x`
C. `5x-1`
D. `10x`

Explanation

To find one of the factors of the expression \( (25x^2 - 1) + (1 + 5x)^2 \), we will follow these steps:

### Step 1: Expand the expression
First, we need to expand the expression \( (1 + 5x)^2 \).

\[ (1 + 5x)^2 = 1^2 + 2 \cdot 1 \cdot 5x + (5x)^2 = 1 + 10x + 25x^2 \]

### Step 2: Substitute back into the original expression
Now, substitute this back into the original expression:

\[ (25x^2 - 1) + (1 + 10x + 25x^2) \]

### Step 3: Combine like terms
Combine the like terms in the expression:

\[ 25x^2 + 25x^2 - 1 + 1 + 10x = 50x^2 + 10x \]

### Step 4: Factor out the common term
Now, we can factor out the common term from \( 50x^2 + 10x \):

\[ 50x^2 + 10x = 10x(5x + 1) \]

### Conclusion
Thus, one of the factors of the expression \( (25x^2 - 1) + (1 + 5x)^2 \) is \( 10x \).