Approach A.
Note that $Z=(1-X)/4\sim U[0,1/4]$, hence by using the moment generating function for (continues) uniform distribution we have that
$$M_Z(t)=\frac{4(e^{t/4}-1)}{t}.$$
Approach B. $$M_Z(t)=\mathbb{E}(e^{tZ})=\mathbb{E}\left(e^{t(1-X)/4}\right)=\mathbb{E}(e^{t/4})+\mathbb{E}\left(e^{-tX/4}\right).$$ Since $-X/4\sim U[-1/4,0]$ we have that $$M_Z(t)=e^{t/4}+\frac{4\left(1-e^{-t/4}\right)}{t}.$$
As there is one unique generating function I guess the second approach is wrong, it would be great if you may help me to spot the mistake.
The second approach should have $e^{t/4} E[e^{-tX/4}]$ which would lead you to the same expression as in the first approach.