<Question>
Let $f(x)$ be the polynomial \[f(x)=3x^4+5x^2-9x-2.\] If $g(x)$ is equal to the polynomial $f(x-1)$, what is the sum of the coefficients of $g$?
</Question>

<think>
<Plan_1>
Okay, let's see. The problem is about polynomials. We have f(x) given as 3x⁴ + 5x² - 9x - 2. Then g(x) is defined as f(x - 1). We need to find the sum of the coefficients of g(x). Hmm, right.

First, I need to recall what the sum of the coefficients of a polynomial means. If I remember correctly, the sum of the coefficients of a polynomial g(x) is found by evaluating g(1). Because when you substitute x = 1 into the polynomial, each term becomes just the coefficient multiplied by 1 raised to some power, which is 1, so you add up all the coefficients.

<prompt_1.1> Recall what the sum of the coefficients of a polynomial means and how to calculate it. </prompt_1.1>

But since g(x) is f(x - 1), then g(1) would be f(1 - 1) = f(0).

Wait, that seems straightforward. So instead of expanding g(x) and adding up its coefficients, which would take time, maybe I can just evaluate f at 0?

Let me check that again. If g(x) = f(x - 1), then substituting x = 1 into g gives g(1) = f(1 - 1) = f(0).
Therefore, the sum of the coefficients of g is f(0). So all I need to do is compute f(0).

<prompt_1.2> Verify the reasoning and decide whether to proceed with evaluating f(0) or expanding g(x). </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
So, for example, if g(x) = 2x³ + 4x² - 5x + 7, then the sum of coefficients is 2 + 4 + (-5) + 7 = 8, and indeed, g(1) = 2(1)³ + 4(1)² - 5(1) + 7 = 2 + 4 - 5 + 7 = 8.
</execution_1.1>

<execution_1.2>
That seems much simpler than expanding the polynomial. Let me verify if this approach is correct.

Let me recall the definition. If g(x) = f(x - 1), then g(x) is the polynomial obtained by shifting f(x) one unit to the right. However, the sum of the coefficients is indeed g(1), which is f(1 - 1) = f(0). So yes, that's correct.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
So let's compute f(0).

<prompt_2.1> Calculate f(0) to find the sum of the coefficients of g(x). </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Given f(x) = 3x⁴ + 5x² - 9x - 2, substituting x = 0 gives f(0) = 3*(0)^4 + 5*(0)^2 - 9*(0) - 2 = 0 + 0 - 0 - 2 = -2. Therefore, the sum of the coefficients of g(x) is -2.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, that seems too easy. Let me make sure I didn't skip any steps. Let's confirm.

<prompt_3.1> Confirm the solution by re-evaluating the logic and calculations. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
The sum of coefficients of g(x) is g(1). Since g(x) = f(x - 1), then g(1) = f(1 - 1) = f(0). Then f(0) is indeed -2 as calculated. So the answer should be -2.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Alternatively, to check, maybe I can compute g(x) by expanding f(x - 1) and then add up the coefficients. Let's try that as a verification.

<prompt_4.1> Expand f(x - 1) to verify the solution. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
Given f(x) = 3x⁴ + 5x² -9x -2. Then f(x - 1) = 3(x - 1)^4 + 5(x - 1)^2 -9(x - 1) -2. Let's expand each term step by step.

First, expand (x - 1)^4. Let's recall that (x - 1)^4 = x^4 - 4x³ + 6x² -4x +1. So multiplying by 3: 3(x -1)^4 = 3x⁴ -12x³ +18x² -12x +3.

Next, expand 5(x -1)^2. (x -1)^2 = x² -2x +1. Multiply by 5: 5x² -10x +5.

Then, expand -9(x -1): -9x +9.

And the constant term is -2.

Now, combine all these terms:

3x⁴ -12x³ +18x² -12x +3

+5x² -10x +5

-9x +9

-2

Combine like terms:

- x⁴ term: 3x⁴

- x³ term: -12x³

- x² terms: 18x² +5x² = 23x²

- x terms: -12x -10x -9x = (-12 -10 -9)x = -31x

- constants: 3 +5 +9 -2 = (3 +5) + (9 -2) = 8 +7 = 15

So g(x) = 3x⁴ -12x³ +23x² -31x +15. Now, the sum of coefficients is 3 + (-12) +23 + (-31) +15. Let's compute that:

3 -12 = -9

-9 +23 = 14

14 -31 = -17

-17 +15 = -2.

So that matches the previous result. Therefore, the sum of the coefficients of g(x) is indeed -2.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the sum of the coefficients of $g$ is $\boxed{-2}$.
</Final_answer>
</think>