Every odd number can be written in a 2k+1 form where k is integer.

Thus for the two consecutive positive odd integers we can write: 2k+1 and 2k+3

the product of this two numbers is 143 that is

(2k + 1)(2k + 3) = 143

4k^2 + 5k - 140 = 0

by solving this quadratic equation we find

k1 = 5

k2 = -7 it is not a solution since we need k >0.

For k = 5 the consecutive positive odd integers are:

2*5 + 1 = 11

2*5 + 3 = 13

and indeed 11*13 = 143

The numbers that we are looking for are 11 and 13.