Solve the equation exactly in the complex number system. x^3+4x^2+9x+36=0

Factor by grouping.
  0 = (x^3 +4x^2) +(9x +36) = x^2(x+4) +9(x+4) = (x^2+9)(x+4)

The first factor is the difference of squares (in complex numbers), so the form for factoring that can be used:
  a^2 - b^2 = (a -b)(a +b)
Here, a=x, b=3i.

Then the factorization in complex numbers is
  0 = (x -3i)(x +3i)(x +4)
Solutions are the values of x that make these factors be zero.

  x = -4, -3i, 3i