To expand the expression $8a(a - 3b + 6c)$, we distribute $8a$ to each term inside the parentheses:

$8a(a - 3b + 6c) = 8a \cdot a - 8a \cdot 3b + 8a \cdot 6c$

Now simplify each term:

1. $8a \cdot a = 8a^2$
2. $-8a \cdot 3b = -24ab$
3. $8a \cdot 6c = 48ac$

So, the expanded expression is:

$8a^2 - 24ab + 48ac$

Final Answer:

$8a^2 - 24ab + 48ac$

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Would you like me to explain any step in detail? Here are five related questions you might find interesting:

1. How do you factorize an expanded polynomial like this?
2. What are common mistakes to avoid when distributing terms in an expression?
3. How do you handle similar expansions with negative coefficients outside the parentheses?
4. How does this process change if the coefficients involve fractions?
5. Can you apply this to expressions with more variables, like $8a(a - 3b + 6c - 2d)$?

Tip: Always double-check the signs when distributing negative terms!