#### Answer

$51$

#### Work Step by Step

We start with the given expression: $3+6(x-2)^3$
We substitute in the given value for $x$: $3+6(4-2)^3$
The order of operations states that first we perform operations inside grouping symbols, such as parentheses, brackets, and fraction bars. Then, we simplify powers. Then, we multiply and divide from left to right. Then, we add and subtract from left to right. We follow the order of operations to simplify:
$$3+6(2)^3$$$$3+6(8)$$$$3+48$$$$\boxed{51}$$