The paper investigates the unconventional superconductivity in the iron-based high-$T_c$ superconductor LaFeAsO$_{1-x}$F$_x$. The authors construct a minimal model that includes all five Fe $d$ bands and apply the random-phase approximation (RPA) to solve the Eliashberg equation. They find that multiple spin fluctuation modes arising from the nesting across disconnected Fermi surfaces lead to an extended s-wave pairing, while d-wave pairing is also a possible candidate. The Fermi surface consists of multiple pockets, and the spin fluctuations dominate over orbital fluctuations when the Hubbard parameter $U$ is greater than $U'$. The gap function shows a sign change between the inner hole and outer electron Fermi pockets, consistent with previous studies. The off-diagonal elements of the gap matrix are significant, indicating a variation in the gap magnitude along the $\beta$ Fermi surface. The study also explores the possibility of d-wave pairing when the $\alpha$ Fermi surfaces are absent or less effective. Future research directions include considering self-energy corrections and comparing the model with experimental data on specific heat and point-contact conductance.The paper investigates the unconventional superconductivity in the iron-based high-$T_c$ superconductor LaFeAsO$_{1-x}$F$_x$. The authors construct a minimal model that includes all five Fe $d$ bands and apply the random-phase approximation (RPA) to solve the Eliashberg equation. They find that multiple spin fluctuation modes arising from the nesting across disconnected Fermi surfaces lead to an extended s-wave pairing, while d-wave pairing is also a possible candidate. The Fermi surface consists of multiple pockets, and the spin fluctuations dominate over orbital fluctuations when the Hubbard parameter $U$ is greater than $U'$. The gap function shows a sign change between the inner hole and outer electron Fermi pockets, consistent with previous studies. The off-diagonal elements of the gap matrix are significant, indicating a variation in the gap magnitude along the $\beta$ Fermi surface. The study also explores the possibility of d-wave pairing when the $\alpha$ Fermi surfaces are absent or less effective. Future research directions include considering self-energy corrections and comparing the model with experimental data on specific heat and point-contact conductance.