I am no mathematician but have studied mathematics some 20 years ago. So I know basics of number theory but have lost the skills to solve problems.
I was wondering if the equation $3ax^2 + (3a^2+6ac)x-c^3=0$ in which $a$ and $c$ are arbitrary positive integers and $c$ is divisible by 6 has an integer solution.
So we need to show there is either no $a$, $c$ and $x$ all positive that satisfy the equation, or one counter example exist.
Rational root theorem didn't help me.