The question is to identify the distribution of the random variable given the moment-generation function. Is there an easy trick to do this or do you have to derive the moment-generating function for every distribution to figure it out?
Here are the three examples:
m(t)=(1-4t)^(-2)
m(t)=1/(1-3.2t)
m(t)=e^(-5t+6t^2)
There is no "trick" to identifying distributions from moment-generating functions besides computing the moment-generating function of the random variable for which you are trying to identify the distribution...then, once you have done this, there are many tables of moment-generating functions for standard distributions, so that you can compare parameters to identify the distribution. The following is a link for a table that you can use: http://www.stat.tamu.edu/~twehrly/611/distab.pdf $$$$ Additionally, recall the definition of the moment-generating function $M_X$ of a random variable $X$ is $$M_X(t) := \mathbb{E}\left[\text{exp}(tX)\right]\,.$$