Hi Irakli,
If R(x) does not have vertical asymptotes, it means you have to “remove” both (x-1) and (x-3) from
R(x), meaning R(x) contains both (x-1) and (x-3) as factors, that is, there is a hole at x=1, and x=3 (instead of having vertical asymptotes at x=1, x=3).
You need to expand (x-1)(x-3) to x^2-4x+3, then using long division to divide R(x) by (x^2-4x+3) to find out what values of b and c will be to ensure remainder in the Long division is 0.
Please let me know if you have further questions.