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