The product of two roots of $x^4-18x^3+kx^2+200x-1984$ is $-32$; what is $k$? 
In the quartic equation, $x^4-18x^3+kx^2+200x-1984$, The product of two roots is $-32$. What is the value of $k$?

This is my effort: Without losing generality, let's say $ab=-32$. Then since $abcd=-1984$, $cd=62$. $bcd+acd+abd+abc=ab(c+d)+cd(a+b)$, so $-32(c+d)+62(a+b)=-200$. Since $a+b+c+d=18$, let's say $a+b=X$, and $c+d=18-X$. $62X-32(18-X)=-200$, $94X+576=-200$, $94X=-776$, $a+b=\frac{776}{94}$

Good attempt!
$$94X\color{red}-576=-200$$
$$X=\frac{376}{94}=4$$
That is now, we know $a+b=4$ and $c+d=18-4=14$.
Also $ab=-32$ and $cd=62$.
While we can solve for the roots, we can observe that we can make use of some tricks:
\begin{align}k&=ab+ac+ad+bc+bd+cd \\&=(ab+cd) +  (ac+ad+bc+bd)
\\
&=(ab+cd) + (a+b)(c+d)\\&=(-32+62)+4(14)\\
&=30+56\\&=86 \end{align}
