  Solving a system of polynomial equations is a ubiquitous problem in the
applications of mathematics. Until recently, it has been hopeless to find
explicit solutions to such systems, and mathematics has instead developed deep
and powerful theories about the solutions to polynomial equations. Enumerative
Geometry is concerned with counting the number of solutions when the
polynomials come from a geometric situation and Intersection Theory gives
methods to accomplish the enumeration.
  We use Macaulay 2 to investigate some problems from enumerative geometry,
illustrating some applications of symbolic computation to this important
problem of solving systems of polynomial equations. Besides enumerating
solutions to the resulting polynomial systems, which include overdetermined,
deficient, and improper systems, we address the important question of real
solutions to these geometric problems.
  The text contains evaluated Macaulay 2 code to illuminate the discussion.
This is a chapter in the forthcoming book "Computations in Algebraic Geometry
with Macaulay 2", edited by D. Eisenbud, D. Grayson, M. Stillman, and B.
Sturmfels. While this chapter is largely expository, the results in the last
section concerning lines tangent to quadrics are new.
