Example 9 Solve the equation
$$
(6 x+7)^{2}(3 x+4)(x+1)=6 .
$$
Solution: Multiply both sides of the original equation by 12, we get
$$
(6 x+7)^{2}(6 x+8)(6 x+6)=72 \text {. }
$$

Let $m=(6 x+7)^{2}, n=(6 x+8)\left(6 x^{2}+6\right)$ $=(6 x+7)^{2}-1$, then
$$
m+(-n)=1, m(-n)=-72 \text {. }
$$

By the inverse of Vieta's theorem, $m$ and $-n$ are the two roots of the equation $y^{2}-y-72=0$. Solving this equation, we get $y=-8$ or $y=9$.
$$
\therefore m=-8 \text { or } m=9 \text {, }
$$

i.e., $(6 x+7)^{2}=-8$ (discard) or $(6 x+7)^{2}=9$.
$$
\therefore x_{1}=-\frac{2}{3}, x_{2}=-\frac{5}{3} \text {. }
$$