We are tasked with simplifying the expression:

$-2g(0) + 3h(-1)$

To solve this, we need to gather the values of $g(0)$ from the provided table and calculate $h(-1)$ using the given equation for $h(x)$.

Step 1: Find $g(0)$

From the table of values for $g(x)$, we see that:

$g(0) = 3$

Step 2: Find $h(-1)$

The function $h(x)$ is given as:

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

Substitute $x = -1$ into the equation for $h(x)$:

$h(-1) = 3(-1)^2 + 2(-1) - 3$ $h(-1) = 3(1) - 2 - 3 = 3 - 2 - 3 = -2$

Step 3: Simplify the expression

Now substitute $g(0) = 3$ and $h(-1) = -2$ into the original expression:

$-2g(0) + 3h(-1) = -2(3) + 3(-2)$ $= -6 + (-6) = -12$

Thus, the simplified form of the expression is:

$\boxed{-12}$

Would you like further clarification on any part of this solution?

Here are some related questions to deepen your understanding:

1. How would the result change if $g(0)$ were 5 instead of 3?
2. Can you compute $g(8)$ and $h(0)$ using the provided information?
3. How does the value of $h(-1)$ change if the coefficient of $x^2$ in $h(x)$ is altered?
4. What would happen if the signs in the expression were reversed, i.e., $2g(0) - 3h(-1)$?
5. Can you create a new function by combining $g(x)$ and $h(x)$ in a different way?

Tip: When solving expressions involving functions, it's helpful to first evaluate each function individually before simplifying the entire expression.