After loading to_poly_solve,
(%i4) to_poly_solve([2*(x-1)*%e^(t*x)+t*(x-1)^2*%e^(t*x)=0,4*t*(x-1)*%e^(t*x)+t^2*(x-1)^2*%e^(t*x)+2*%e^(t*x)>0],[x]); (%o4) %union([x = 1,2 > 0],[x = (t-2)/t, t^2-2*(t-2)*t-4*t+4*(t-2)+(t-2)^2+2 > 0])
I haven't really used it much for inequalities before, but note that the 2>0 should just remove this solution. Also,
(%o2) - 2
so I'm not sure whether the conditional piece is really necessary at all here?
First reported at http://ask.sagemath.org/question/2045/solve-contains-wrong-solution with a bit of a challenge from Mathematica output in one of the answers, for some reason...