Example 6 If $x^{2}+x+2 m$ is a perfect square, then $m=$ $\qquad$
$(1989$, Wu Yang Cup Junior High School Mathematics Competition)
Solution: If $x^{2}+x+2 m$ is a perfect square, then the quadratic equation in $x$, $x^{2}+x+2 m=0$, has equal roots. Thus,
$$
\Delta=1-4 \times(2 m)=0 .
$$

Solving for $m$ yields $m=\frac{1}{8}$.
Explanation: According to the sufficient condition for a real-coefficient quadratic polynomial $a x^{2}+b x+c(a \neq 0)$ to be a perfect square, which is $c>0$ and $b^{2}-4 a c=0$, the value of the parameter in the perfect square can be determined.