To solve this problem, we need to substitute $x = -6$ into the function given:

$f(x) = -x^6 + 5x^4 + 8x^3 + 2x - 7$

Substituting $x = -6$ into each term:

1. Calculate $-(-6)^6$: $(-6)^6 = 46656, \quad \text{so} \quad -(-6)^6 = -46656$

2. Calculate $5(-6)^4$: $(-6)^4 = 1296, \quad \text{so} \quad 5 \cdot 1296 = 6480$

3. Calculate $8(-6)^3$: $(-6)^3 = -216, \quad \text{so} \quad 8 \cdot (-216) = -1728$

4. Calculate $2(-6)$: $2 \cdot (-6) = -12$

5. The constant term is $-7$.

Now add all the terms together:

$f(-6) = -46656 + 6480 - 1728 - 12 - 7$

Let's simplify step-by-step:

$-46656 + 6480 = -40176$ $-40176 - 1728 = -41904$ $-41904 - 12 = -41916$ $-41916 - 7 = -41923$

So, $f(-6) = -41923$.

The correct answer is:

$f(-6) = -41923$

Would you like a detailed breakdown of each step, or do you have any further questions?

Related Questions:

1. How do you evaluate functions with multiple terms when substituting values?
2. What are some common mistakes to avoid when substituting negative values in polynomial functions?
3. How can powers of negative numbers be simplified?
4. What role do each of the function's terms play in determining its value at specific points?
5. How would the result change if $x$ was a positive value instead?

Tip:

When substituting values in a polynomial, carefully calculate each power separately, especially when working with negative bases, as they can significantly affect the result.