Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define:
$$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \mathbb{R^3}\}.$$ Then:
(a) $D$ is connected (as a subset of the topological space $\mathbb{R}^3$).
(b) For any $(a,b,c)\in D$, we have $a^2\geq 3b$.
This was a true/false question in one of my test (NBHM 2023). I have found some conditions but those were seems to be quite overwhelming. Can someone please give me some hint? My guess is it may be the case that $D$ is path connected.