The product of two of the four roots of the polynomial

$$
P(X)=X^{4}-18 X^{3}+k X^{2}+200 X-1984
$$

is equal to -32. Find $k$.
Let $a$ and $a'$ be the roots of $P$ such that their product is $-32$, and let $b$ and $b'$ be the other two roots of $P$. We know that $aa'bb' = -1984$, so $bb' = \frac{-1984}{-32} = 62$. Thus, $a' = -\frac{32}{a}$ and $b' = \frac{62}{b}$. Furthermore, $a + a' + b + b' = a - \frac{32}{a} + b + \frac{62}{b} = 18$, while $aa'bb'\left(\frac{1}{a} + \frac{1}{a'} + \frac{1}{b} + \frac{1}{b'}\right) = -1984\left(\frac{1}{a} - \frac{a}{32} + \frac{1}{b} + \frac{b}{62}\right) = -200$, or equivalently, $-\frac{a + a'}{32} + \frac{b + b'}{62} = -\frac{200}{1984}$. These two equations indicate that $a + a' = 4$ and $b + b' = 14$. Since $k = aa' + bb' + (a + a')(b + b')$, it follows that $k = 86$, which, conversely, is indeed suitable.