Here is my solution.

A = 2^x

y = A^2 - 5A = A(A - 5) = k (A > 0)

Now plot y = A^2 - 5A (quadratic)

Clearly, for two solutions for A > 0, A must be between 0 and 5. By sketching, it is clear maximum is 0 (not inclusive) for two solutions, and the minimum of the function (minimum of quadratic) is -25/4 (not inclusive).

Hence, k is in (-25/4, 0)