1. Given that $x$ is an irrational number, and $(x+1)(x+3)$ is a rational number. Under the above assumptions, there are the following four conclusions:
(1) $x^{2}$ is a rational number;
(2) $(x-1)(x-3)$ is an irrational number;
(3) $(x+1)^{2}$ is a rational number;
(4) $(x-1)^{2}$ is an irrational number.
Among them, the number of correct conclusions is ( ).
(A) 0
(B) 1
(C) 2
(D) 3
- 1. C.

Take the irrational number $x=\sqrt{3}-2$, at this time,
$$
(x+1)(x+3)=(\sqrt{3}-1)(\sqrt{3}+1)=2
$$

is a rational number.
While $x^{2}=(\sqrt{3}-2)^{2}=7-4 \sqrt{3}$ is an irrational number, hence conclusion (1) is incorrect;

By $(x+1)^{2}=(\sqrt{3}-1)^{2}=4-2 \sqrt{3}$ is an irrational number, hence conclusion (3) is incorrect.
$$
\begin{array}{l}
\text { By }(x-1)(x-3)=x^{2}-4 x+3 \\
=x^{2}+4 x+3-8 x=(x+1)(x+3)-8 x \text {, }
\end{array}
$$
and $(x+1)(x+3)$ is a rational number, $8 x$ is an irrational number, hence $(x-1)(x-3)$ is an irrational number, i.e., conclusion (2) is correct.
Similarly, by
$$
(x-1)^{2}=(x+1)(x+3)-6 x-2 \text {, }
$$

we know that conclusion (4) is correct.