Solution
= Solution
At zero, $F'(0)=-(1+\mu)$, so it is stable for $-2<\mu<0$, loses stability through multiplier $+1$ at $-2$ and $-1$ at $0$. Nonzero fixed points are
$$
x_\pm=\frac{-q\pm\sqrt{q^2+4p(2+\mu)}}{2p},
$$
born in a saddle-node at $\mu=-2-q^2/(4p)$. Their stability follows from $F'(x_*)=3+\mu+px_*^2$; $|F'|<1$ marks solid branches. Thus the three bifurcations are the saddle-node, the exchange at $-2$, and the flip at $0$.