1. 58 Let $x$ be an irrational number, but $(x-2)(x+6)$ is a rational number, then which of the following conclusions is correct?
(A) $x^{2}$ is a rational number.
(B) $(x+6)^{2}$ is a rational number.
(C) $(x+2)(x-6)$ is an irrational number.
(D) $(x+2)^{2}$ is an irrational number.
(China Sichuan Province Junior High School Mathematics Competition, 1989)
$$
\text { [Solution] From } \begin{aligned}
& (x+2)(x-6)=x^{2}-4 x-12 \\
= & (x+6)(x-2)-8 x,
\end{aligned}
$$

If $(x-2)(x+6)$ is a rational number, and $8 x$ is an irrational number, then $(x+2)(x-6)$ is an irrational number.
Therefore, the answer is $(C)$.