Graphing both y=5^(3*x+2) and y = 1-x shows one point of intersection, and it's between x=-1 and x=0.

Program:

  x = linspace(-1,0,500);

  y1 = 5.^(3.*x+2);

  y2=1-x;

  figure

  plot(x,y1 )

  grid

  hold on

  plot(x,y2)

  axis equal

  

  

  low=-1; high=   0 ; 

  

for iter=1:30

  prev=0;

for x=low:(high-low)/10:high

  v=x+5^(3*x+2)-1;

  % disp([x v])

  if sign(v)~=sign(prev) && prev~=0

    break

  end

  prev=v;

  prevx=x;

end

low=prevx; high=x;

disp([x v])

if prev==v

  break

end 

end

first plots the graph, then (added after looking at the graph) homes in on the solution, x=-0.572865929453477, or rather in the puzzle's nomenclature N = -0.572865929453477.