To factor the expression \( a^2b^2 + a \) completely, we can look for common factors in the terms.
Observe that both terms \( a^2b^2 \) and \( a \) contain a factor of \( a \).
We can factor out \( a \):
\[ a^2b^2 + a = a(a b^2 + 1) \]
Now, we have factored the expression as much as possible. Therefore, the completely factored form of \( a^2b^2 + a \) is:
\[ \boxed{a (ab^2 + 1)} \]