im not sure your liklihood is correct.
The likelihood is proportional to
this shouldn't affect your final answer as long as you did your algebra correctly (the constant term at the front vanishes anyway).
To show your final answer is unbiased, evaluate:
You know X is binomial and so E(X) = 0.5n(1+t).
Im assuming that you are allowed to treat the moments of a binomial distribution as standard results instead of having to derive them every time