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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}

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}

or FindRoot providing good initial guesses,

FindRoot[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t]}, {t, #}] & /@ {-5, -3, 1, 4} 

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

Using 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}

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}

FindRoot needs a good initial guess.

Plot[{Abs[3 - t^2] + Abs[t - 1] - t^2, 2*Sin[t]}, {t, -8, 8}] 

Experimenting with the initial guess,

FindRoot[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 2*Sin[t]}, {t, #}] & /@ {0, 6} 

{{t -> 0.885124}, {t -> 3.43039}}

Using 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}