To solve the expression $h(-3) \cdot g(4)$, we'll first extract the given functions $h(x)$ and $g(x)$, and then evaluate them at $x = -3$ and $x = 4$, respectively.

Step 1: Identify the functions

Based on the information provided:

• $h(x)$ is given by the following set of values: \begin{cases} 33 & \text{if } x = -8 \\ -3 & \text{if } x = 2 \\ -4 & \text{if } x = -4 \\ 7 & \text{if } x = -3 \\ 6 & \text{if } x = 7 \end{cases}$$
• $g(x) = 3x^2 - 2x + 2$

Step 2: Find $h(-3)$

From the table of values for $h(x)$, we see that: $h(-3) = 7$

Step 3: Find $g(4)$

Now, substitute $x = 4$ into the function $g(x)$: $g(4) = 3(4)^2 - 2(4) + 2$ First, calculate the powers and products: $g(4) = 3(16) - 8 + 2 = 48 - 8 + 2 = 42$

Step 4: Multiply $h(-3)$ and $g(4)$

Now, multiply the results: $h(-3) \cdot g(4) = 7 \cdot 42 = 294$

Thus, the simplified form of $h(-3) \cdot g(4)$ is $294$.

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Would you like more details on any step, or do you have further questions?

Here are five related questions:

1. How do we handle different forms of function compositions?
2. What are the steps to simplify quadratic functions?
3. Can we use other methods to check values for piecewise functions like $h(x)$?
4. What is the role of squaring in quadratic functions?
5. How can we efficiently compute the product of two function values?

Tip: Always substitute values carefully, especially when dealing with polynomials and piecewise functions, to avoid mistakes!