To evaluate the function $f(x) = 3x^2 - 5x + 1$ at $x = -2$, substitute $-2$ for $x$ in the function:

$f(-2) = 3(-2)^2 - 5(-2) + 1$

Now, calculate step by step:

1. Calculate $(-2)^2$: $(-2)^2 = 4$

2. Multiply by 3: $3 \times 4 = 12$

3. Calculate $-5 \times (-2)$: $-5 \times (-2) = 10$

4. Add the results together: $12 + 10 + 1 = 23$

Therefore, $f(-2) = 23$.

If you have any further questions or would like additional details, feel free to ask!

Here are some related questions for further exploration:

1. What is the value of $f(x)$ when $x = 0$?
2. How do you find the vertex of the quadratic function $f(x) = 3x^2 - 5x + 1$?
3. How can you determine if this quadratic function opens upwards or downwards?
4. What are the roots of the function $f(x) = 3x^2 - 5x + 1$?
5. How would you find the axis of symmetry for the function?
6. How do you complete the square to rewrite the function in vertex form?
7. What is the y-intercept of the function $f(x) = 3x^2 - 5x + 1$?
8. How can you graph the function $f(x) = 3x^2 - 5x + 1$ using key features like intercepts and vertex?

Tip: When evaluating a function, carefully substitute values and perform arithmetic operations step by step to avoid errors.