There are two possibilities to consider, depending on which way 
subtracting them gives 2:

Case 1.  The roots are  and , or 
Case 2.  The roots are  and , or 

In case 1, the difference between the roots is 
In case 2, the difference between the roots is 

In case 1, the solutions would be

 and 

or

 and 

or

 and 

which would have come from using the zero-factor 
property of the factored equation:



which would have come from the quadratic equation:



or

 

But the problem said that the coefficients are positive 
integers.  But -56 is not positive.  So we have to discard that answer.

---

In case 2, the solutions would be

 and 

or

 and 

or

 and 

which would have come from using the zero-factor 
property of the factored equation:



which would have come from the quadratic equation:



or

 

The coefficients of  and  are both
positive. So that is the only quadratic equation
with the coefficients of  and  both
positive integers. 

Edwein