$29 \cdot 45$ If $x^{2}+2 x+5$ is a factor of $x^{4}+p x^{2}+q$, then the values of $p$ and $q$ should be
(A) $-2,5$.
(B) 5,25 .
(C) 10,20 .
(D) 6,25 .
(E) 14,25 .
(11th American High School Mathematics Examination, 1960)
[Solution] Using synthetic division, divide $x^{4}+p x^{2}+q$ by $x^{2}+2 x+5$ to get the quotient $x^{2}-2 x+p-1$, and the remainder $(12-2 p) x+q-5 p+5$.
From the problem, we know that $(12-2 p) x+q-5 p+5=0$,
which means $12-2 p=0$ and $q-5 p+5=0$.
Solving these equations, we get
$$
p=6, q=25 \text {. }
$$

Therefore, the answer is $(D)$.