Let's evaluate the function $f(x) = 3x^2 + 2x - 2$ for each part:

(a) $f(0)$

Substitute $x = 0$:

$f(0) = 3(0)^2 + 2(0) - 2 = -2$

(b) $f(4)$

Substitute $x = 4$:

$f(4) = 3(4)^2 + 2(4) - 2 = 3(16) + 8 - 2 = 48 + 8 - 2 = 54$

(c) $f(-4)$

Substitute $x = -4$:

$f(-4) = 3(-4)^2 + 2(-4) - 2 = 3(16) - 8 - 2 = 48 - 8 - 2 = 38$

(d) $f(-x)$

Substitute $x = -x$:

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

(e) $-f(x)$

Multiply the function by $-1$:

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

(f) $f(x+2)$

Substitute $x = x + 2$:

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

Simplify:

$f(x+2) = 3(x^2 + 4x + 4) + 2x + 4 - 2 = 3x^2 + 12x + 12 + 2x + 2 = 3x^2 + 14x + 14$

(g) $f(3x)$

Substitute $x = 3x$:

$f(3x) = 3(3x)^2 + 2(3x) - 2 = 3(9x^2) + 6x - 2 = 27x^2 + 6x - 2$

(h) $f(x+h)$

Substitute $x = x + h$:

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

Simplify:

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

Let me know if you would like more details or have any questions.

Here are some related questions:

1. What are the roots of the function $f(x) = 3x^2 + 2x - 2$?
2. How do you find the vertex of the parabola represented by $f(x) = 3x^2 + 2x - 2$?
3. What is the derivative of $f(x) = 3x^2 + 2x - 2$?
4. How do transformations affect the graph of $f(x) = 3x^2 + 2x - 2$?
5. What are the asymptotes of a rational function derived from $f(x) = 3x^2 + 2x - 2$?

Tip: Always check for simplification opportunities when substituting into functions, as they can often reduce computation errors and make the problem easier to solve.