I don't know its roots. I only know the product of 2 of its roots

Let the four roots be $a$ , $b$ , $c$ and $d$ . Using Vieta's formula , we have:

$\begin{cases} a+b+c+d = 18 &...(1) \\ ab+ac+ad+bc+bd+cd = k &...(2) \\ abc+abd+acd+bcd = -200 &...(3) \\ abcd = -1984 &...(4) \end{cases}$

Let $ab=-32$ , then from $(4): cd = \dfrac {-1984}{ab} = \dfrac {-1984}{-32} = 62$

From (3):

$\begin{aligned} abc+abd+acd+bcd & = -200 \\ ab(c+d)+cd(a+b) & = -200 \\ -32(c+d)+62(a+b) & = -200 \\ -32({\color{#3D99F6}a+b+c+d})+94 (a+b) & = -200 & \small \color{#3D99F6} (1): a+b+c+d = 18 \\ -32({\color{#3D99F6}18})+94 (a+b) & = -200 \end{aligned}$

$\implies a+b = \dfrac {32\times 18-200} {94} = 4$

From (1): $c+d = 18-(a+b) = 18-4 = 14$

From (2):

$\begin{aligned} k & = ab+ac+ad+bc+bd+cd \\ & = ab + (a+b)(c+d) + cd \\ & = -32 + (4)(14) + 62 \\ & = \boxed{86} \end{aligned}$