jmc

We use the Euclidean Algorithm. $$\begin{array}{rl}\text{gcd}(3a^3+a^2+4a+57,a)& =\text{gcd}(3a^3+a^2+4a+57-(3a^2+a+4)a,a)\\ & =\text{gcd}(57,a).\end{array}$$Since $57$ is a divisor of $456$, and $a$ is a multiple of $456$, the greatest common divisor is $\boxed{57}$.