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You can use NSolve to find multiple roots,

NSolve[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], -5 <= t <= 5}, t] 

{{t -> -4.92954}, {t -> -3.77452}, {t -> 0.762901}, {t -> 3.35745}}

or FindAllCrossings from here,

FindAllCrossings[Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], {t, -10, 10}, WorkingPrecision -> 20] 

{-4.9295434376879868373, -3.7745180124835511931, 0.76290087955924483126, 3.3574487606850113852}