FindRoot[Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], {t, #}] & /@ {-3, 1, 3}(* Starting values from Plot[]*)
(* {{t -> -3.77452}, {t -> 0.762901}, {t -> 3.35745}} *)
Or:
NSolve[{RealAbs[3 - t^2] + RealAbs[t - 1] - t^2 == 3*Sin[t], -4 < t < 4}, t, Reals](*Why Abs[] dosen't work I don't know !.*)
(* {{t -> -3.77452}, {t -> 0.762901}, {t -> 3.35745}} *)
NSolve[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], -4 < t < 4}, t, Complexes]
(* {{t -> -3.77452}, {t -> 0.762901}, {t -> 3.35745}} *)
Solve[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], -4 < t < 4}, t, Reals]
(*{{t -> Root[{-4 - 3 Sin[#1] + #1 &, 3.3574487606850113852}]}, {t ->
Root[{2 + 3 Sin[#1] + #1 &, -3.7745180124835511931}]}, {t ->
Root[{-4 + 3 Sin[#1] + #1 + 2 #1^2 &, 0.76290087955924483126}]}}*)
%//N
(* {{t -> 3.35745}, {t -> -3.77452}, {t -> 0.762901}} *)
Reduce[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], -4 < t < 4}, t, Reals]
(* t == Root[{-4 - 3 Sin[#1] + #1 &, 3.3574487606850113852}] ||
t == Root[{2 + 3 Sin[#1] + #1 &, -3.7745180124835511931}] ||
t == Root[{-4 + 3 Sin[#1] + #1 + 2 #1^2 &, 0.76290087955924483126}] *)
%//N
(* t == 3.35745 || t == -3.77452 || t == 0.762901*)
FindInstance[{Abs[3 - t^2] + Abs[t - 1] - t^2 == 3*Sin[t], -4 < t < 4}, t, Reals, 3] // N
(* {{t -> 3.35745}, {t -> -3.77452}, {t -> 0.762901}} *)