Given the function $f(x) = 3x^2 - x + 2$, let's calculate the values of each requested expression.

1. Find $f(2)$: $f(2) = 3(2)^2 - (2) + 2 = 3(4) - 2 + 2 = 12 - 2 + 2 = 12$

2. Find $f(-2)$: $f(-2) = 3(-2)^2 - (-2) + 2 = 3(4) + 2 + 2 = 12 + 2 + 2 = 16$

3. Find $f(a)$: $f(a) = 3a^2 - a + 2$

4. Find $f(-a)$: $f(-a) = 3(-a)^2 - (-a) + 2 = 3a^2 + a + 2$

5. Find $f(a + 1)$: $f(a + 1) = 3(a + 1)^2 - (a + 1) + 2$ First expand $(a + 1)^2 = a^2 + 2a + 1$: $f(a + 1) = 3(a^2 + 2a + 1) - (a + 1) + 2$ Now simplify: $f(a + 1) = 3a^2 + 6a + 3 - a - 1 + 2 = 3a^2 + 5a + 4$

6. Find $2f(a)$: $2f(a) = 2(3a^2 - a + 2) = 6a^2 - 2a + 4$

7. Find $f(2a)$: $f(2a) = 3(2a)^2 - (2a) + 2 = 3(4a^2) - 2a + 2 = 12a^2 - 2a + 2$

8. Find $f(a^2)$: $f(a^2) = 3(a^2)^2 - a^2 + 2 = 3a^4 - a^2 + 2$

9. Find $[f(a)]^2$: $[f(a)]^2 = (3a^2 - a + 2)^2$ Expand: $[f(a)]^2 = (3a^2)^2 - 2(3a^2)(a) + 2(3a^2)(2) + (-a)^2 - 2(-a)(2) + 2^2$ Simplify: $[f(a)]^2 = 9a^4 - 6a^3 + 12a^2 + a^2 - 4a + 4 = 9a^4 - 6a^3 + 13a^2 - 4a + 4$

10. Find $f(a + h)$: $f(a + h) = 3(a + h)^2 - (a + h) + 2$ Expand $(a + h)^2 = a^2 + 2ah + h^2$: $f(a + h) = 3(a^2 + 2ah + h^2) - (a + h) + 2$ Now simplify: $f(a + h) = 3a^2 + 6ah + 3h^2 - a - h + 2$

Summary of Results:

1. $f(2) = 12$
2. $f(-2) = 16$
3. $f(a) = 3a^2 - a + 2$
4. $f(-a) = 3a^2 + a + 2$
5. $f(a + 1) = 3a^2 + 5a + 4$
6. $2f(a) = 6a^2 - 2a + 4$
7. $f(2a) = 12a^2 - 2a + 2$
8. $f(a^2) = 3a^4 - a^2 + 2$
9. $[f(a)]^2 = 9a^4 - 6a^3 + 13a^2 - 4a + 4$
10. $f(a + h) = 3a^2 + 6ah + 3h^2 - a - h + 2$

Would you like any further details or clarifications on these calculations?

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Follow-up Questions:

1. How do you derive the formula for the square of a binomial, like in the case of $f(a + 1)$?
2. Can you apply these techniques to find the derivative of $f(x)$?
3. What happens if you plug specific values like $a = 1$ or $h = 0$ into these expressions?
4. How would you solve for $a$ given that $f(a) = 0$?
5. What are some other functions of the form $f(x) = ax^2 + bx + c$ and how do they behave differently?

Tip:

When expanding expressions like $(a + h)^2$, it's often helpful to first expand the square term and then simplify step by step. This method avoids mistakes and keeps calculations clear.