To factor the expression \( a^2 + 10a + 25 - 4b^2 \), we can start by identifying the perfect squares and reformatting the expression.

1. Notice that \( a^2 + 10a + 25 \) can be factored as:
\[
(a + 5)^2
\]
because \( (a + 5)(a + 5) = a^2 + 10a + 25 \).

2. The term \( -4b^2 \) can be rewritten as:
\[
- (2b)^2
\]

Putting it all together, the expression becomes:
\[
(a + 5)^2 - (2b)^2
\]

3. Now, we can apply the difference of squares formula, which states that \( x^2 - y^2 = (x - y)(x + y) \).

Here,
- Let \( x = (a + 5) \) and \( y = 2b \).

Thus, we have:
\[
(a + 5 - 2b)(a + 5 + 2b)
\]

So, the factored form of the expression \( a^2 + 10a + 25 - 4b^2 \) is:
\[
(a + 5 - 2b)(a + 5 + 2b)
\]