  A More Sums Than Differences (MSTD) set is a set of integers A contained in
{0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it
is known that as n tends to infinity a positive percentage of subsets of {0,
..., n-1} are MSTD sets, the methods to prove this are probabilistic and do not
yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave
explicit constructions of a large family of MSTD sets; though their density is
less than a positive percentage, their family's density among subsets of {0,
..., n-1} is at least C/n^4 for some C>0, significantly larger than the
previous constructions, which were on the order of 1 / 2^{n/2}. We generalize
their method and explicitly construct a large family of sets A with |A+A+A+A| >
|(A+A)-(A+A)|. The additional sums and differences allow us greater freedom
than in Miller, Orosz and Scheinerman, and we find that for any epsilon>0 the
density of such sets is at least C / n^epsilon. In the course of constructing
such sets we find that for any integer k there is an A such that |A+A+A+A| -
|A+A-A-A| = k, and show that the minimum span of such a set is 30.
