Example 4 Let $A=\left\{x \mid x^{2}+(p+2) x+1=0, x \in \mathbf{R}\right\}$, if $A \varsubsetneqq \mathbf{R}^{-}$, find the range of real number $p$.

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Let $f(x)=x^{2}+(p+2) x+1$. Since $f(0)=1$, to make $A \varsubsetneqq \mathbf{R}^{-}$, the equation $x^{2}+(p+2) x+1=0$ can only have two negative roots or no solution, i.e., $\left\{\begin{array}{l}\Delta=(p+2)^{2}-4 \geqslant 0, \\ -(p+2)<0\end{array}\right.$ or $(p+2)^{2}-4<0$. Solving this, we get $p \geqslant 0$ or $-4<p<0$. Therefore, the range of real numbers $p$ is $(-4,+\infty)$.

Key to solving the problem: Recognize the set language and convert it into natural language that you can understand, and combine it with the graph where all values are negative real numbers, or $A=\varnothing$.