$4 \cdot 53$ Find the product of all real solutions to the equation $x^{2}+25 x+52=3 \sqrt{x^{2}+25 x+80}$.
［Solution］Let $\sqrt{x^{2}+25 x+80}=u$, then the original equation can be transformed into
$$
u^{2}-28=3 u \text {. }
$$

Since $u \geqslant 0$, we have $u=7$.
That is, $x^{2}+25 x+80=7^{2}$.
By $\Delta>0$, we know that the equation has distinct real roots. By Vieta's formulas, the product of the two roots is $80-$ $49=31$.