Your differential equation is of the second order, but you give three initial data. Also, at the the starting point x==0 of the equation is singular:
NDSolve[{T[x] + 4 (-1 + 2 x) Derivative[1][T][x] +
4 (-1 + x) x (T^\[Prime]\[Prime])[x] == 0, T[0] == 1,
T'[0] == 1/4, T''[0] == 9/32}, T, {x,
0, 1}]
DSolve gives a solution, which is probably equivalent with yours:
DSolve[Prepend[ICs, Eq1], T, x]
