8. For the quartic polynomial $x^{4}-18 x^{3}+k x^{2}+200 x-1984$, the product of two of its four zeros is -32, then the real number $k=$
Rong Ge 86.
Analysis Let the four roots of the polynomial $x^{4}-18 x^{3}+k x^{2}+200 x-1984$ be $x_{1}, x_{2}, x_{3}, x_{4}$, then by Vieta's formulas, we get
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+x_{3}+x_{4}=18 \\
x_{1} x_{2}+x_{1} x_{3}+x_{1} x_{4}+x_{2} x_{3}+x_{2} x_{4}+x_{3} x_{4}=k \\
x_{1} x_{2} x_{3}+x_{1} x_{2} x_{4}+x_{1} x_{3} x_{4}+x_{2} x_{3} x_{4}=-200 \\
x_{1} x_{2} x_{3} x_{4}=-1984
\end{array}\right.
$$

Let $x_{1} x_{2}=-32$, then $x_{3} x_{4}=62$, so $62\left(x_{1}+x_{2}\right)-32\left(x_{3}+x_{4}\right)=-200$;
Also, $x_{1}+x_{2}+x_{3}+x_{4}=18$, so $\left\{\begin{array}{l}x_{1}+x_{2}=4 \\ x_{3}+x_{4}=14\end{array}\right.$, hence $k=x_{1} x_{2}+x_{3} x_{4}+\left(x_{1}+x_{2}\right)\left(x_{3}+x_{4}\right)=86$.