Let $S_c := \{(x,y,z) \in \mathbb{R}^3 \,| \,\, z(z+4) = 3xy + c \}$. Find all values of $c\in\mathbb{R}$ such that:
$1) \,\, S_c$ is a regular surface in $\mathbb{R}^3$
$2) \,\, S_c$ is connected
$3) \,\, S_c$ is compact
For part $1)$, I used the implicit function theorem: let $F(x,y,z):= z^2 + 4z-3xy-c$, so $S_c = F^{-1}(\{0\})$. Then $\nabla F=0$ for $p=(0,0,-2)$, and p is in $S_c$ for $c=-4$.
So, for $c \neq -4, \,\, S_c$ is a regular surface. If $c=-4$ the implicit function theorem cannot say if $S_{-4}$ is singular or not. So, to check, we study
$S_{-4}=\{(z+2)^2=3xy\}$ I'm not sure how to see if this is a regular surface. I tried parametrizing it as a graph: $$\phi(x,y)=(x,y,\pm \sqrt{3xy} - 2)$$ and then calculating $\phi_x = (1,0,\pm \frac{3y}{2\sqrt{3xy}})$ and $\phi_y = (0,1,\pm \frac{3x}{2\sqrt{3xy}})$ I observed that these are linearly independent for all $x,y$ and concluded that $S_c$ is a regular surface for all $c$. Is this line of reasoning correct? Something doesn't convince me.
For part $3)$, I would say that $S_c$ is not compact for any $c$ since for every $c$ I can always arbitrarily increase $z$ and find appropriate $x,y$ to satisfy the condition on $S_c$, so the set is not limited. How do I put this in a formal way?
I have no idea how to approach the connected part. How do I tell if something is connected by looking at an implicit function definition? Is there a general way to attack these problems?
I'm hoping someone else will weigh in on a general approach to deduce connectedness. In your example, with the change of variables $u=x+y$, $v=x-y$, you have
$$ (z+2)^2+(3/4)v^2-(3/4)u^2 = c+4 $$
Going by the classification of quadric surfaces this is either a one-sheet hyperboloid or two-sheet hyperboloid depending on whether $c+4$ is positive or negative.