Find all positive values for k for which the following
can be factored.

x²+ x - k

It can be factored if the discriminant B²-4AC is the
square of an integer. A=1, B=1, C=-k

B²-4AC = 1²-4(1)(-k) = 1+4k

We see that 1+4k is an odd number, and can be any odd
perfect square.  But any odd perfect square is the
square of an odd integer. 

Any positive odd number can be represented by 2n+1 
where n is a non-negative integer, so

1 + 4k = (2n+1)²

1 + 4k = 4n² + 4n + 1

    4k = 4n² + 4n
     k = n² + n

     k = n(n+1) for any non-negative integer.

But since k can't be 0, we have to rule out n=0 
and change it to

    k = n(n+1) for any POSITIVE integer n.

So k must be any term of this sequence 2, 6, 12, 20, 30, ... 

Edwin