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Let's simplify and factor the given expression:

\[
12x^2y - 32xy + 8y
\]

### Step 1: Factor out the greatest common factor (GCF)
The common factor in all terms is \( 4y \), so we can factor that out:

\[
4y(3x^2 - 8x + 2)
\]

### Step 2: Factor the quadratic expression \( 3x^2 - 8x + 2 \)
Now, we need to factor the quadratic expression inside the parentheses: \( 3x^2 - 8x + 2 \). To factor this, we look for two numbers that multiply to \( 3 \times 2 = 6 \) and add to \( -8 \). These numbers are \( -6 \) and \( -2 \).

Next, split the middle term:

\[
3x^2 - 6x - 2x + 2
\]

### Step 3: Group and factor by grouping
Now group the terms and factor each group:

\[
(3x^2 - 6x) - (2x - 2)
\]

Factor out the greatest common factors from each group:

\[
3x(x - 2) - 1(x - 2)
\]

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

\[
(3x - 1)(x - 2)
\]

### Final factored form:
The fully factored expression is:

\[
4y(3x - 1)(x - 2)
\]

Would you like further explanation or details on any of the steps?

### Related Questions:
1. How do you identify the greatest common factor in an expression?
2. What is the process for factoring a quadratic trinomial?
3. How does grouping work when factoring by parts?
4. Why is it important to check the factored form by expanding it again?
5. Can all quadratic expressions be factored, or are there special cases?

### Tip:
When factoring quadratics, always start by checking for a greatest common factor (GCF) before applying other factoring methods.

Write this expression in its most simplified, factored form: 12x^2y - 32xy + 8y

Learn how to simplify and factor the expression 12x^2y - 32xy + 8y. This step-by-step solution explains the process of factoring the greatest common factor (GCF) and factoring a quadratic expression using grouping techniques. Suitable for students in Grades 9-12.

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