$30 \cdot 28$ If $x$ is a natural number, let $y=x^{4}+2 x^{3}+2 x^{2}+2 x+1$, then
(A) $y$ must be a perfect square.
(B) There are a finite number of $x$ such that $y$ is a perfect square.
(C) $y$ must not be a perfect square.
(D) There are infinitely many $x$ such that $y$ is a perfect square.
(China Partial Provinces and Cities Junior High School Mathematics Correspondence Competition, 1990)
[Solution] Let $y=\left(x^{4}+2 x^{2}+1\right)+\left(2 x^{3}+2 x\right)$
$=\left(x^{2}+1\right)^{2}+2 x\left(x^{2}+1\right)$
$=\left(x^{2}+1\right)(x+1)^{2}$,
Since $(x+1)^{2}$ is a perfect square, but $x^{2}+1$ is not a perfect square, then $\left(x^{2}+1\right)(x+1)^{2}$ is not a perfect square.
Therefore, the answer is $(C)$.