To factor the expression \( 25c^2 - 49 \), we recognize that it is a difference of squares. The general form for a difference of squares is:
\[ a^2 - b^2 = (a - b)(a + b) \]
In this case, we can identify \( a = 5c \) and \( b = 7 \), since:
\[ 25c^2 = (5c)^2 \quad \text{and} \quad 49 = 7^2 \]
Now we can rewrite the expression:
\[ 25c^2 - 49 = (5c)^2 - 7^2 \]
Using the difference of squares formula, we factor it as follows:
\[ (5c - 7)(5c + 7) \]
Thus, the completely factored form of the expression \( 25c^2 - 49 \) is:
\[ \boxed{(5c - 7)(5c + 7)} \]