The product of two consecutive positive odd integers is 255. what are the integers?

Let the smaller odd positive integer be n.

Then

n (n+2) = 255

solve by expanding to quadratic equation n^2+2n-255=0, reject negative root, or

by trial and error: sqrt (255) = 15.97 = 16, so the odd integers should be 15, 17.

Check: 15*17=255 ... correct!

Alternatively, let the even number sandwiched between the odd integers be x.

Then

(x+1) (x-1) = 255

x^2-1=255

x^2=255+1=256

x=sqrt (256) = 16

so the required odd integers are 16-1=15, and 16+1=17.

Your choice!