A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find zn, the maximum number of regions into which n zig-zags can divide the plane. For example, z1=2,z2=12(see the diagram). Of these zn regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in n of degree not exceeding 2.
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy] algebrapolynomialcombinatorics unsolvedcombinatorics