The relation between addition and multiplication in R[X] and in R[X]/(d) can be expressed using the reduction map (or residue class map)
r : R[X] -> R[X]/(d),
f -> f + (d)
(reduction modulo d; the map sends every polynomial to its equivalence class). The definitions for sum and product of residue classes express that
r(a) + r(b) = r(a + b) and r(a)r(b) = r(ab).