1. **State the problem:** Given the function $f(x) = 3x^2 - x + 2$, we need to find the values of $f(2)$, $f(-2)$, $f(a)$, $f(-a)$, $f(a+1)$, $2f(a)$, $f(2a)$, $f(a^2)$, $[f(a)]^2$, and $f(a+h)$. 2. **Find $f(2)$:** Substitute $x = 2$ into the function. $$f(2) = 3(2)^2 - (2) + 2 = 3 \times 4 - 2 + 2 = 12 - 2 + 2 = 12$$ 3. **Find $f(-2)$:** Substitute $x = -2$. $$f(-2) = 3(-2)^2 - (-2) + 2 = 3 \times 4 + 2 + 2 = 12 + 2 + 2 = 16$$ 4. **Find $f(a)$:** Substitute $x = a$. $$f(a) = 3a^2 - a + 2$$ 5. **Find $f(-a)$:** Substitute $x = -a$. $$f(-a) = 3(-a)^2 - (-a) + 2 = 3a^2 + a + 2$$ 6. **Find $f(a+1)$:** Substitute $x = a+1$. $$f(a+1) = 3(a+1)^2 - (a+1) + 2 = 3(a^2 + 2a + 1) - a - 1 + 2 = 3a^2 + 6a + 3 - a - 1 + 2 = 3a^2 + 5a + 4$$ 7. **Find $2f(a)$:** Double the expression for $f(a)$. $$2f(a) = 2(3a^2 - a + 2) = 6a^2 - 2a + 4$$ 8. **Find $f(2a)$:** Substitute $x = 2a$. $$f(2a) = 3(2a)^2 - (2a) + 2 = 3(4a^2) - 2a + 2 = 12a^2 - 2a + 2$$ 9. **Find $f(a^2)$:** Substitute $x = a^2$. $$f(a^2) = 3(a^2)^2 - a^2 + 2 = 3a^4 - a^2 + 2$$ 10. **Find $[f(a)]^2$:** Square the expression for $f(a)$. $$[f(a)]^2 = (3a^2 - a + 2)^2$$ This expands to: $$= (3a^2)^2 - 2 \times 3a^2 \times a + 2 \times 3a^2 \times 2 + a^2 - 2 \times a \times 2 + 2^2 = 9a^4 - 6a^3 + 12a^2 - 4a + 4$$ (Alternatively, it can be left as is for simplicity.) 11. **Find $f(a+h)$:** Substitute $x = a + h$. $$f(a+h) = 3(a+h)^2 - (a+h) + 2 = 3(a^2 + 2ah + h^2) - a - h + 2 = 3a^2 + 6ah + 3h^2 - a - h + 2$$ **Final answers:** $f(2) = 12$, $f(-2) = 16$, $f(a) = 3a^2 - a + 2$, $f(-a) = 3a^2 + a + 2$, $f(a+1) = 3a^2 + 5a + 4$, $2f(a) = 6a^2 - 2a + 4$, $f(2a) = 12a^2 - 2a + 2$, $f(a^2) = 3a^4 - a^2 + 2$, $[f(a)]^2 = (3a^2 - a + 2)^2$, $f(a+h) = 3a^2 + 6ah + 3h^2 - a - h + 2$