5.32 $f(x)$ is a function defined on all real numbers, and its graph is symmetric about $x=2$. It is known that when $x \in(-2,2]$ the expression of $f(x)$ is $-x^{2}+1$. Then when $x \in(-6,-2)$, the expression of $f(x)$ is
(A) $-x^{2}+1$.
(B) $-(x-2)^{2}+1$.
(C) $-(x+2)^{2}+1$.
(D) $-(x+4)^{2}+1$.
(China Beijing High School Mathematics Competition, 1990)
[Solution] First, draw the graph of $y=-x^{2}+1$ on $(-2,2]$. Since the graph of $y=f(x)$ is symmetric about $x=2$, we can then draw the function graph on $[2,6)$, with the equation being
$$
y=-(x-4)^{2}+1 \text {, }
$$

Furthermore, since $y=f(x)$ is an even function, meaning its graph is symmetric about the $y$-axis (as shown in the figure), the expression for $f(x)$ on $(-6, -2)$ is
$$
-(x+4)^{2}+1 .
$$

Therefore, the correct choice is $(D)$.