$4 \cdot 64$ Find the product of the real roots of the equation $x^{2}+18 x+30=2 \sqrt{x^{2}+18 x+45}$
[Solution] Let $y=\sqrt{x^{2}+18 x+45}$. From the original equation, we get
$$y^{2}-15=2 y,$$

Rearranging gives $y^{2}-2 y-15=0$,
Solving yields $y_{1}=5, y_{2}=-3$ (discard $y<0$).
Thus, we have
$$\sqrt{x^{2}+18 x+45}=5$$

which simplifies to
$$x^{2}+18 x+20=0$$

By the discriminant $\Delta=18^{2}-4 \times 20>0$, we know the equation has two real roots. By Vieta's formulas, the product of these two real roots is 20, i.e., the product of the real roots of the given equation is 20.