Obviously, this doesn't hold in general. I was watching this video and at 11:40 the creator claims that if $g = fh$ and $g$ is inseparable, while $f$ is separable, then $h$ must not be separable. This doesn't seem to hold at all (consider the field $\mathbb{F}_2(y^2)$, where $y$ is transcendental then $g(x) = x^2+y^2$ is inseparable but $g(x) = (x+y) \cdot (x+y)$). It's difficult to transcribe the entire content of the video, but I'll try to include the important parts.
The author is trying to prove that separability is a transitive property, i.e. if $A \subseteq B \subseteq C$ and $C/B$ and $B/A$ are separable, then so is $C/A$. He defines $S \subseteq C$ as the set of all elements of $C$ separable over $A$. He proves $S$ is a field over which $C$ is separable, which I know and am not confused about. Then, he takes an element $\alpha \in C \setminus S$ and defines $f$ as its minimal polynomial over $S$, $g$ as its minimal polynomial over $A$. By assumption, $f$ is separable, while $g$ is not. Also, $f \mid g$ as $f$ is minimal over $S$ and $g \in A[x] \subseteq S[x]$. This is where he writes $g = fh$ and concludes $h$ is nonseparable.
Can you help me understand why this is?