Let the zeros be $a, b, c, d$. The relationship between zeros and coefficients yields \begin{align} a + b + c + d &= 18\\ ab + ac + ad + bc + bd + cd & = k\\ abc + abd + acd + bcd &= -200\\ abcd & = -1984 \end{align} Assume $ab = -32$ and let $u = a + b, v = c + d, w = cd$. Then \begin{align} u + v & = 18\\ -32 + uv + w &= k\\ -32v + uw &= -200\\ -32w &= -1984 \end{align} From the last equation we get $w = 62$, and replacing it in the other equations leads to $u = 4, v = 14$. Hence $$k = -32 + 4 \times 14 + 62 = \boxed{86}$$