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In this work, we propose a management method for controlling the speed and direction of self-bound quantum droplets (QDs) in a binary Bose-Einstein condensate mixture under time-modulated external harmonic confinement. Utilizing the 1D extended Gross-Pit"{a}evskii equation, QDs are constructed within both regular and expulsive parabolic traps, considering temporally varying attractive quadratic beyond mean field and repulsive cubic mean-field atom-atom interactions. Through the derived wavefunction solution, we illustrate the dynamics of slowing, stopping, reversing, fragmentation, collapse, and revival of droplets. Additionally, the solutions reveal a crystalline order with a superfluid background, indicative of supersolid behavior in various parameter domains. Notably, one-third of the constant background matches the lowest residual condensate. These findings hold potential applications in matter-wave interferometry and quantum information processing.