Example $1 . a$ For what value of $a$, 
$$
(a+1) x^{2}+(a-3) x+(a-5)=0
$$
Given: Taking $n=1$, $m=3$, then
$$
\begin{array}{c}
(a-3)^{2}-4(a+1)(a-5)=9(a+1)^{2}, \\
a=1 \text { or } -\frac{5}{3} .
\end{array}
$$