Let's use algebra to solve this problem.
Let x be one of the integers, and y be the other integer.
From the problem, we know that:
x = 4y + 17 (one integer is 17 more than 4 times the other)
xy = -15 (the product of the two integers is -15)
We can use substitution to solve for one of the variables. From the first equation, we can write:
y = (x - 17)/4
Substituting this expression for y into the second equation gives:
x(x - 17)/4 = -15
Multiplying both sides by 4 and expanding gives:
x^2 - 17x = -60
Rearranging gives:
x^2 - 17x + 60 = 0
This is a quadratic equation that can be factored as:
(x - 5)(x - 12) = 0
So the solutions are:
x = 5, y = (-12/4) = -3
x = 12, y = (-5/4) = -1.25
Since the problem states that the integers are both integers, the only valid solution is:
x = 5, y = -3
Therefore, the two integers are 5 and -3.