= Solution
A group is <solvable group>[solvable] when its <derived series>
$$
G^{(0)}=G,\qquad G^{(r+1)}=[G^{(r)},G^{(r)}]
$$
eventually reaches the trivial group; equivalently, it has a subnormal series with <abelian group>[abelian] quotients.
The <symmetric group> $S_4$ is solvable because
$$
\{1\}\triangleleft V_4\triangleleft A_4\triangleleft S_4
$$
has abelian quotients $V_4$, $C_3$, and $C_2$, where $V_4$ is the <Klein four-group>. Thus
$$
\boxed{S_4\text{ is solvable}}.
$$
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