Let $R$ be a non-zero ring with identity. Is it possible for $R[x]$ to have only a finite number of maximal left ideals !?
Lets try something like Euclid's proof of infiniteness of prime numbers. Let $f_1 = x, f_2 = xf_1 + 1, f_3=xf_1f_2+1, f_4=xf_1f_2f_3 + 1$ and so on. The left ideal generated by each $f_i$ is proper. Thus each $f_i$ is contained in a maximal left ideal. On the other hand for any $i < j$ we have $ f_i | f_j-1$ (note that $f_i$'s are in the center of $R[x]$). This means that $f_i$ and $f_j$ can not be both in one maximal left ideal. Hence we have an infinite number of maximal left ideals.