Let x be the first odd integer and x + 2 be the second intger.

\(x(x+2) = 1023\)

\(x^2 +2x = 1023\)

\(x^2 +2x -1023 =0 \)

\((x+33)(x-31) =0 \)

\(x_1 =-33 \)

\(x_2 =31 \)

The first integer is 31 and the second integer is 33.