Given that a is an integer, find possible $a$'s so that $2^{2a+1}+2^a+1$ is a perfect square

Question

I found this question in an old Olympiad from my country, but the solution I found only provides a brute force approach.

The solution said $0$ and $4$, and when it reaches the point it tests $8$, the solution stops there.

I wonder if there is another way to solve it?