To simplify the expression \((a+b)^3 - 8c^3\), we will use the binomial theorem for the expansion of \((a+b)^3\) and the formula for the difference of cubes.

First, expand \((a+b)^3\):

\[
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]

Next, rewrite the expression \((a+b)^3 - 8c^3\):

\[
(a+b)^3 - 8c^3 = (a^3 + 3a^2b + 3ab^2 + b^3) - (2c)^3
\]

Recognize that \(8c^3 = (2c)^3\), and we can use the formula for the difference of cubes:

\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]

Here, \(x = a+b\) and \(y = 2c\). Applying the difference of cubes formula:

\[
(a+b)^3 - (2c)^3 = ((a+b) - 2c)((a+b)^2 + (a+b)(2c) + (2c)^2)
\]

Now, let's simplify each part:

1. \((a+b) - 2c = a + b - 2c\)
2. \((a+b)^2 = a^2 + 2ab + b^2\)
3. \((a+b)(2c) = 2c(a + b) = 2ac + 2bc\)
4. \((2c)^2 = 4c^2\)

Combine these:

\[
(a+b)^3 - (2c)^3 = (a + b - 2c) \left( a^2 + 2ab + b^2 + 2ac + 2bc + 4c^2 \right)
\]

Therefore, the simplified form is:

\[
(a+b)^3 - 8c^3 = (a + b - 2c) \left( a^2 + 2ab + b^2 + 2ac + 2bc + 4c^2 \right)
\]

Would you like more details or have any questions?

Here are some questions you might want to ask next:
1. How do I apply the binomial theorem to higher powers?
2. Can you explain the steps of simplifying a different polynomial expression?
3. How do you factor trinomials?
4. What are some examples of using the difference of squares formula?
5. How do you solve polynomial equations?

**Tip:** When dealing with polynomial expansions, remembering and applying fundamental formulas like the binomial theorem and the difference of cubes can significantly simplify the process.

Learn how to simplify the expression (a+b)^3 - 8c^3 using the binomial theorem for expansion and the difference of cubes formula. This problem involves algebraic manipulation and polynomial simplification techniques. Suitable for high school students.

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