Let's evaluate $f(x) = 3x^2 - 2$ at each specific value and simplify $f(x+1)$.

1. Evaluating $f(x+1)$:

We are asked to find $f(x+1)$, which means we substitute $x+1$ into the function wherever $x$ appears:

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

First, expand $(x+1)^2$:

$(x+1)^2 = x^2 + 2x + 1$

Now substitute this into the function:

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

Distribute the 3:

$f(x+1) = 3x^2 + 6x + 3 - 2$

Simplify:

$f(x+1) = 3x^2 + 6x + 1$

So, $f(x+1) = 3x^2 + 6x + 1$.

Do you have any questions or need further details?

Here are five related questions to expand on this topic:

1. What happens if you evaluate $f(-1)$ in the same function?
2. How would you approach solving $f(2)$ for this function?
3. Can you simplify $f(x-1)$ instead of $f(x+1)$?
4. How would the graph of $f(x) = 3x^2 - 2$ look?
5. What are the roots of the equation $f(x) = 0$?

Tip: When evaluating functions at shifted values like $f(x+1)$, always substitute carefully and simplify step by step to avoid mistakes.