Example 5 The inequality $x^{2}+|2 x-4| \geqslant p$ holds for all real numbers $x$. Find the maximum value of the real number $p$.

保留了原文的换行和格式，如上所示。
Explanation 1: Transform the original inequality into
$|x-2| \geqslant-\frac{x^{2}}{2}+\frac{p}{2}$.
Let $y_{1}=|x-2|, y_{2}=-\frac{x^{2}}{2}+\frac{p}{2}$.
Thus, solving inequality (1) is equivalent to determining the range of $x$ values for which the graph of the function $y_{1}=|x-2|$ is above the graph of the function $y_{2}=-\frac{x^{2}}{2}+\frac{p}{2}$. In this problem, we are given that this range is all real numbers, and we need to find the maximum value of the parameter $p$.

As shown in Figure 5,
draw the graphs of
the functions $y_{1}=|x-2|$
and $y_{2}=-\frac{x^{2}}{2}$
$+\frac{p}{2}$.
It is easy to see that the critical case where the graph of $y_{1}=|x-2|$ is above the graph of $y_{2}=-\frac{x^{2}}{2}+\frac{p}{2}$ is when $p=3$. Therefore, the maximum value of $p$ is 3.

Explanation 2: Let $y=x^{2}+|2 x-4|$. We only need to draw the graph of the function $y=x^{2}+|2 x-4|$.
When $x \geqslant 2$,
$$
y=x^{2}+2 x-4 \text {; }
$$

When $x<2$,
$$
y=x^{2}-2 x+4 \text {. }
$$

As shown in Figure 6, it is clear that the maximum value of $p$ is 3.

Note: Using function graphs to combine numerical and geometric insights is another effective approach to discussing parameters, such as the number of real roots of an equation or the solution set of an inequality. It is important to note that although the graphs are rough sketches, the key conditions, key points, and trends must be accurately represented, otherwise, there will be misjudgments.