Solve the inequality
\[\frac{x^2 - 25}{x + 5} < 0.\]
We can factor the numerator, to get
\[\frac{(x - 5)(x + 5)}{x + 5} < 0.\]If $x \neq -5,$ then this simplifies to $x - 5 < 0.$  Since the expression is not defined for $x = -5,$ the solution is
\[x \in \boxed{(-\infty,-5) \cup (-5,5)}.\]