  The main theme of this dissertation is retooling methods to work for
different situations.
  I have taken the method derived by O'Hara and simplified by Zeilberger to
prove unimodality of $q$-binomials and tweaked it. This allows us to create
many more families of polynomials for which unimodality is not, a priori,
given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer
and Zeilberger proved a result about bounded lattice walks. I employ their
generating function relation technique to analyze lattice walks with a general
step set in bounded, semi-bounded, and unbounded planes. The method in which we
do this is formulated to be highly algorithmic so that a computer can automate
most, if not all, of the work. I easily recover many well-known results for
simpler step sets and discover new results for more complex step sets.
