Consider the following cubic equation in $c$:
$c^3 - 3c^2(a+b) + 3c(a+b) -3ab(a+b)-3=0$
Does this equation have infinitely many integer solutions $(a,b,c)$ ?
EDIT: My attempt was rerwriting it as a linear in $a+b$, such that $a+b=\dfrac{c^3 -3}{3c^2 +3ab- 3c}$
Which shows that $a+b$ is an integral solution whenever $3c^2 + 3ab -3c$ divides $c^3 -3$. The question now would be on the existence of infinitely many integers $(a, b, c)$ such that the divisibility holds ?