We first find $f(-2)$ by substituting $-2$ into the function $f(x)$:
$f(-2) = (-2)^3 + 3 = -8 + 3 = -5$.
Next, we find $g(f(-2))$ by substituting $-5$ into the function $g(x)$:
$g(f(-2)) = 2(-5)^2 + 2(-5) + 1 = 2(25) - 10 + 1 = 50 - 10 + 1 = \boxed{41}$.
The answer is: 41
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Given the functions $f(x) = x^3 + 3$ and $g(x) = 2x^2 + 2x + 1$, what is the value of $g(f(-2))$?

### Answer:
