I am looking for a general solution to
$$(2a^2)^2+(ab)^2+4=c^2$$ where $a,b,c, \in\mathbb{Z^+}$
In fact, it is possible to find the values that work. But, I am looking for a general solution. At least, I want to know if there are infinitely many solutions to this equation.
My effort is only research and brute-force. I found that this equation looks like Pythagorean quadruple.But, I have no idea what kind of method can be applied to the equation above. And I don't know if this equation can be solved by elementary number theory.I don't know anything about the Method. So, it's pointless to talk about my steps.
Taking $(a,b)=(n,n^3)$ for any $n\in\mathbb{N}$, we have $$(2n^2)^2+(n\cdot n^3)^2+4=4n^4+n^8+4=(n^4+2)^2$$ So we have infinitely many solutions given by $$(a,b,c)=(n,n^3,n^4+2)\quad\forall n\in\mathbb{N}$$