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We are given the polynomial \( 12c^2 - 38c + 20 \), and we need to determine its factors. The task is to completely factor this quadratic expression and identify the correct options from the list provided.

### Step 1: Factor the polynomial.

First, let's factor \( 12c^2 - 38c + 20 \) using the method of factoring quadratic expressions by trial and error or grouping.

We are looking for two numbers that multiply to give the product of the coefficient of \( c^2 \) and the constant term (i.e., \( 12 \times 20 = 240 \)) and add up to the middle term's coefficient (i.e., \( -38 \)).

We need two numbers that multiply to 240 and add to -38. These numbers are -30 and -8.

So, we can split the middle term:

\[
12c^2 - 30c - 8c + 20
\]

### Step 2: Group and factor by grouping.

Now, group the terms in pairs:

\[
(12c^2 - 30c) - (8c - 20)
\]

Factor each group:

\[
6c(2c - 5) - 4(2c - 5)
\]

### Step 3: Factor the common binomial.

Now, factor out the common binomial factor \( (2c - 5) \):

\[
(6c - 4)(2c - 5)
\]

### Step 4: Factor out the greatest common factor (GCF) from \( 6c - 4 \).

We can factor out the GCF of 2 from \( 6c - 4 \):

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

### Step 5: Conclusion

The completely factored form of the polynomial is:

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

### Step 6: Check the correct options from the list.

From the provided list, the factors of \( 12c^2 - 38c + 20 \) are:

- \( 2 \) (Option A)
- \( 3c - 2 \) (Option C)
- \( 2c - 5 \) (Option D)

Thus, the correct answers are **A, C, and D**.

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Do you want more details on the factoring steps or have any further questions? Here are some related questions you might find useful:

1. How do you factor quadratic polynomials using the quadratic formula?
2. What is the difference between factoring by grouping and trial and error?
3. Can all quadratics be factored over real numbers?
4. What is the importance of factoring in solving quadratic equations?
5. How does the discriminant help in determining the nature of the roots?

**Tip:** Always check for a greatest common factor (GCF) before attempting to factor a quadratic polynomial, as it simplifies the process!

When the polynomial below is completely factored, what are its factors? 12c^2 - 38c + 20. Mark all of the polynomial's factors.

Discover the steps to factor the quadratic polynomial 12c^2 - 38c + 20. Learn about factoring by grouping and using the ac method, suitable for high school students studying algebra.

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