Let's break down the problem you provided step by step. We are asked to find the value of:

$-5h(-6) + 4g(-1)$

We are also given the following information:

1. A table of values for $f(x)$ (though it may not be directly relevant to $h(x)$ and $g(x)$ unless specified):

   • $f(4) = 5$
   • $f(7) = 6$
   • $f(0) = -9$
   • $f(-8) = 4$

2. The function $g(x)$ is defined as: $g(x) = -x^2 - 5x - 1$

However, the function $h(x)$ is not directly defined. To proceed, let's assume $h(x)$ is related to $f(x)$, and we can interpret it using the table, where values of $h(x)$ correspond to values of $f(x)$.

Step 1: Evaluate $h(-6)$

Since $h(x)$ isn't explicitly defined, if $h(x) = f(x)$, we check if $f(-6)$ is in the table, but it's not provided. Without more details, we can't evaluate $h(-6)$ definitively.

Step 2: Evaluate $g(-1)$

We can evaluate $g(-1)$ using the given formula for $g(x)$: $g(-1) = -(-1)^2 - 5(-1) - 1$ Simplify: $g(-1) = -1 + 5 - 1 = 3$

Step 3: Substitute the values back into the expression

We have: $-5h(-6) + 4g(-1) = -5h(-6) + 4(3)$ This simplifies to: $-5h(-6) + 12$

Since we don't have a clear definition for $h(-6)$, we can't simplify this expression further. If $h(x) = f(x)$, and $f(-6)$ is missing from the provided table, we'd need that value to fully simplify.

Would you like further clarification or more details on any part of the problem?

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Here are 5 related questions to think about:

1. What is the role of function $h(x)$ in this problem?
2. How would you define a function based on a table of values?
3. What happens when a function's values aren't given for certain inputs?
4. Can we infer more about $f(x)$ from the table or create an equation for it?
5. How do you graph quadratic functions like $g(x) = -x^2 - 5x - 1$?

Tip: Always make sure that all functions used in a problem are clearly defined before attempting to solve it.