find all positive values for k, if it can be factored

Allowed values of k for for the expression to be factorable with integers are
where a is an integer equal to 1 or more.
For example: 2, 6, 12, 20...
The factors are
There are also real non-integer values of k that allow factrs, as long as 1+4k >0

To determine the allowed values of k for the expression to be factorable, we need to find the quadratic factors of the expression x^2 + x - k.

First, let's rewrite the quadratic expression in the form of (x - a)(x + b), where a and b are integers.

Expanding this form, we have:
x^2 + x - k = (x - a)(x + b)

To find a and b, we can compare the coefficients of the original expression and the expanded form.

Comparing the x^2 terms:
1 (coefficient of x^2) = 1 (coefficient of x in (x - a)(x + b))

Comparing the x terms:
1 (coefficient of x) = -a + b (sum of coefficients of x)

Comparing the constant terms:
- k (constant term in the original expression) = -a * b (product of the constant terms)

From the first comparison, we see that the coefficient of x^2 is the same in both expressions. This tells us that a must be equal to 1.

Substituting a = 1 into the second equation, we get:
1 = -1 + b
b = 2

So, the quadratic expression x^2 + x - k can be factored as:
(x - 1)(x + 2)

To find the allowed values of k, we need to consider integers a and b. In this case, a = 1 and b = 2.

The allowed values of k are given by the product of a and b:
k = 1 * 2 = 2

Therefore, the only positive value for k that allows the expression x^2 + x - k to be factored with integers is k = 2.

Note: In addition to the above, there may also be real non-integer values of k that allow factoring, as long as 1 + 4k > 0.