if p and q are both prime numbers, x^2 - (8p - 10q)x +5pq = 0 has two positive integer solutions

hey, thank you so much! would it be okay if you described the process you went through to get those answers?

Use the quadratic formula to get the roots in terms of p and q.

Being integers and primes and non-negative put constraints on the resulting discriminant (what's under the radical sign)

Solving for values that meet those constraints gets you the two pairs.