Question

how many integer values are there for K for which 4X squared plus KXY minus 9Y squared is factorable

Answer 1

4x^2 + kxy - 9y^2

k = (4x^2 + 9y^2)/xy

let x = 3m and y = 2m

k = (36m^2 + 36n^2)/6mn
k = 6m/n + 6n/m

So, we see that n must divide 6m and m must divide 6n

If m=1, n divides 6: 1,2,3,6
If m=2, n divides 12: 1,2,3,4,6,12
and so on

Now, how many of those values of m and m determine unique values for k, I don't know. Have fun!