Example 12 (from the 1st American Invitational Mathematics Examination) Find the product of the real roots of the equation $x^{2}+18 x+30=2 \sqrt{x^{2}+18 x+45}$.
Let $\sqrt{x^{2}+18 x+45}=y$, then the original equation transforms into $y^{2}-15=2 y$, which is $y^{2}-2 y-15=0$.
Solving for $y_{1}=5, y_{2}=-3$ (discard), thus $\sqrt{x^{2}+18 x+45}=5$, which leads to $x^{2}+18 x+20=0$.
By $\Delta=18^{2}-4 \cdot 20>0$, we know the above equation has two real roots.
By Vieta's formulas, we know the product of the two real roots is 20.