Answer
See below.
Work Step by Step
(a) $(x-r)(x-s)=x^2-(r+s)x+rs$
i) both $r$ and $s$ are odd integers, we have $-(r+s)$ as an even integer and $rs$ as odd integer.
ii) both $r$ and $s$ are even integers, we have $-(r+s)$ as an even integer and $rs$ as even integer.
iii) one even and one odd integers, we have $-(r+s)$ as an odd integer and $rs$ as even integer.
(b) Answers from part-(a) covers all the possible cases of $r$ and $s$, for $x^2-1253x+255$, coefficients $-(r+s)$ and $rs$ are both odd, which is not a case in the results from part-(a), thus it cannot be written as a product of two polynomials with integer coefficients.
