Example 13 The solution to the equation $\frac{1}{x^{2}+x}+\frac{1}{x^{2}+3 x+2}+$ $\frac{1}{x^{2}+5 x+6}+\frac{1}{x^{2}+7 x+12}=\frac{4}{21}$ is $\qquad$ -
(Tenth Zu Chongzhi Cup Junior High School Mathematics Competition)