The product of two positive integers is 120 one number is 2 less than five times the other

x, y - two positive integers

product of two positive integers is 120

(1) xy = 120

one number is 2 less than five times the other

(2) x = 5y - 2 substitute it to (1)

(5y - 2) y = 120 use distributive property

(5y) (y) - (2) (y) = 120

5y² - 2y = 120 subtract 120 from both sides

5y² - 2y - 120 = 0

use quadratic formula

a = 5, b = - 2, c = - 120

b² - 4ac = (-2) ^2 - 4 (5) (-120) = 4 + 2400 = 2404

√2404 is irrational number, therefore

Answer: no solution in integer numbers