I don't quite undestand what you are trying to do. The following
xc1 = Solve[x''''[t] + 5 x''[t] + 4 - Sin[t] - Cos[t] == C[1], x[t]]
gives an empty set because it is not a differential equation and x[t] does not appear in the equation.
To give specific values to the constants you can use a replacement rule:
ecuación = x''''[t] + 5 x''[t] + 4 == Sin[t] + Cos[2 t]
xg = DSolveValue[ecuación, x[t], t]
xg /. {C[1] -> 0, C[2] -> 0, C[3] -> 0, C[4] -> 0}
