The expression \( 12c^2 - 24c - 15 \) can be factored by finding the greatest common factor (GCF) first, and then checking if it can be factored further.

1. **Find the GCF**:
   The GCF of the terms \( 12c^2 \), \( -24c \), and \( -15 \) is 3. So, we factor out 3:

   \[
   12c^2 - 24c - 15 = 3(4c^2 - 8c - 5)
   \]

2. **Factor the quadratic inside the parentheses**:
   We now focus on factoring the quadratic \( 4c^2 - 8c - 5 \). To factor this, we need two numbers that multiply to \( 4 \times -5 = -20 \) and add to \( -8 \).

   The two numbers that work are \( -10 \) and \( 2 \), so we can split the middle term:

   \[
   4c^2 - 10c + 2c - 5
   \]

3. **Group terms**:
   Now, group the terms:

   \[
   (4c^2 - 10c) + (2c - 5)
   \]

4. **Factor by grouping**:
   Factor out the common factors from each group:

   \[
   2c(2c - 5) + 1(2c - 5)
   \]

5. **Final factorization**:
   Factor out the common binomial factor \( (2c - 5) \):

   \[
   3(2c + 1)(2c - 5)
   \]

So, the factored form of \( 12c^2 - 24c - 15 \) is:

\[
\boxed{3(2c + 1)(2c - 5)}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you factor quadratics using the splitting-the-middle-term method?
2. Can you always factor quadratic expressions? If not, when is it not possible?
3. What is the quadratic formula and how does it relate to factoring?
4. How do you solve a quadratic equation by factoring?
5. What is the difference between factoring and expanding an expression?

**Tip**: Always check if the quadratic expression can be factored before using the quadratic formula—it saves time!

Discover how to factor the quadratic expression 12c^2 - 24c - 15 using the Greatest Common Factor and grouping method. This detailed step-by-step guide is perfect for students in Grades 8-10 and covers essential algebraic concepts.

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