The problem presented asks to determine $g(g(x))$ given two functions:

1. $f(x) = x^2 - 3x + 2$
2. $g(x) = 4x + 1$

To solve $g(g(x))$, we substitute the function $g(x)$ into itself:

$g(g(x)) = g(4x + 1)$

Now, using the definition of $g(x)$, replace $x$ in $g(x)$ with $4x + 1$:

$g(4x + 1) = 4(4x + 1) + 1$

Expand the expression:

$g(4x + 1) = 16x + 4 + 1 = 16x + 5$

Thus, $g(g(x)) = 16x + 5$.

Would you like further details or explanations?

Here are five related questions to explore:

1. What is $f(g(x))$ if $f(x) = x^2 - 3x + 2$ and $g(x) = 4x + 1$?
2. How would you find $g(f(x))$ with the same functions?
3. How can we determine the inverse of the function $g(x) = 4x + 1$?
4. What is the derivative of $g(g(x))$?
5. How would you solve for $x$ if $g(g(x)) = 0$?

Tip: For function composition, always substitute the inner function first before simplifying.