The solution of the puzzle is given below.

Since the product of two of the roots of the given equation is -32; it follows that, the product of the remaining roots is equal to -1984/ -32 = 62. Accordingly, we can write the given quartic expression as the product of two quadratic expressions, so that:

x^4 – 18x^3 + K*x^2 + 200x – 1984

= (x^2 + px -32)(x^2 + qx + 62);(say)

Comparing the exponents of x, x^2 and x^3 in the above relationship, we obtain:

(I) p+q = -18; (II) K = 30 + pq; and (III) 62*p – 32*q = 200;

Solving for p and q in (I) and (III), we obtain, (p,q) = (-4,-14); giving :

K= 30 + (-4)*(-14) = 86.

Note: For a harder variation of quartic methodologies, refer to "A Quartic Problem"

( Reference: *http://perplexus.info/show.php?pid=4279*).

*Edited on ***May 15, 2006, 12:56 pm**

*Edited on ***May 15, 2006, 1:00 pm**