expanding out the left side, $$(t-c)(t-\bar{c})=t^2-ct-\bar{c}t+c\bar{c}=t^2-(c+\bar{c})t+|c|^2,$$ but it is not generally true that $c+\bar{c}=c\bar{c}$. To answer your other question: in general, if $R$ is a ring (an abstract kind of "number system" where we can add and multiply), then $R[x]$ denotes the collection of polynomials in the variable $x$ having coefficients