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.