text: any other solution would have a longer path. And calculating the maximum sum for these two paths is rather trivial ...
  For instance, is it legal to try to do a tour of the matrix, to pick up as many high value cells as possible? If the priority is "max sum", followed by "min terms", then a "rook's tour" of the positive valued cells is in order.
  Another idea, can you revisit a cell? What's to keep you from circling through 4 positive value cells forever?
  What does slide mean? If you slide A->D, skipping B and C, what is the path length?3 or 1? Also, does sliding mean that the points on the squares you slided over are not counted to your sum? Only the final square you landed on is counted?
  Of course this would require cleanly defined rules and not these "oh good point I forgot to mention that ..." dialogs.°
  In short, no need to brute force all possible paths if there are plenty optimal ones and picking one is sufficient.
  Of course I might have misunderstood the "rules", but why should I risk to invest more efforts for vain ? ;)
  is it really allowed to slide back a way you just slid forward? (Like your first two moves a0-up9-down3)
  is it allowed to slide over positive weights? (Like your third move -left1 ignoring 3 positive cells in between)
  most probably automatically constructed from smaller segments like: "finish a column/line before switching to the next one"
  > There are certainly many paths meeting that criterion most probably automatically constructed from smaller segments like
  OK not trivial, but well studied and becoming easier with growing number of possible moves (i.e. low number of non-positive cells here)
  Your example has 36 cells with only 11 non positive ones. That means only one third of the cells need to be skipped.
  After excluding start and stop it's obvious that the best theoretical result is to pass by all remaining 24=34-10 positive ones exactly one time, hence in minimal 25 moves.
  tybalt89 has demonstrated that such a path exists, provided his "slides" were legal. (I already expressed my doubt. Please clarify!!! )