Let y = x^2 + 18x.  Then the equation becomes y + 30 = 2*sqrt[y+45].

This simplifies to the quadratic y^2 + 56y + 720 = 0, which has roots -20 and -36.

Checking these roots: y=-20 yields 10=10 but y=-36 yields -6=+6.  So only the roots corresponding to y=-20 are solutions.

Then y=-20 implies x^2 + 18x = -20, or x^2 + 18x + 20 = 0, which implies the product of the roots of the equation is 20, which is the answer being sought.