Answer: 20

Small Hint:

The expression outside the radical is $15$ less than the radicand

Big Hint:

Set $t=\sqrt{x^2+18x+45}$ and solve $t^2-15=2t$

Solution:

Set $t=\sqrt{x^2+18x+45},$ so $t\geq0.$ The equation becomes $$t^2-15=2t,$$ or $(t-5)(t+3)=0.$ Thus $t=5,$ and $$x^2+18x+45=25,$$ so the real roots satisfy $x^2+18x+20=0.$ By Vieta’s formulas, their product is $20.$