  We have phenomenologically investigated the decays $B_s^0 \to X(3872)
\pi^+\pi^- (K^+ K^-)$ and $B_s^0 \to \psi(2S) \pi^+ \pi^- (K^+K^-)$. In our
analysis, the scalar meson $f_0(980)$ is formed through the final state
interactions of coupled channels $\pi \pi$ and $K\bar{K}$. Our findings
indicate that the $\pi^+\pi^-$ invariant mass distribution of the $B_s^0 \to
\psi(2S)\pi^+\pi^-$ decay can be accurately reproduced. Furthermore, we have
explored the $\pi^+\pi^- (K^+ K^-)$ invariant mass distribution of the $B_s^0
\to X(3872) \pi^+\pi^- (K^+ K^-)$ decay, accounting for the different
production mechanisms between $X(3872)$ and $\psi(2S)$, up to a global factor.
It is found that the production rates for $X(3872)$ and $\psi(2S)$ are much
different, which indicates that the structure of $X(3872)$ is more complicated
than the $\psi(2S)$, which is a conventional $c\bar{c}$ state. Additionally, we
have considered the contributions from $f_0(1500)$ to $\pi^+\pi^-$ and the
$\phi$ meson to $K^+ K^-$ in our analysis. Utilizing the model parameters, we
have calculated the branching fraction of $B_s^0 \to X(3872) K^+ K^-$, and
anticipate that the findings of our study can be experimentally tested in the
future.
