https://file.aiquickdraw.com/mathbot/file/1726024114341a6kjdx39.jpg

Let's go through the factoring process for each polynomial provided in the image.

### a) \( 3x^3 - 2x^2 - 12x + 8 \)

This is a cubic polynomial, and it can be factored using **grouping**:
1. Group the terms: 
   \[
   (3x^3 - 2x^2) - (12x - 8)
   \]
2. Factor out common terms from each group:
   \[
   x^2(3x - 2) - 4(3x - 2)
   \]
3. Now, factor out the common binomial factor \( (3x - 2) \):
   \[
   (3x - 2)(x^2 - 4)
   \]
4. Notice that \( x^2 - 4 \) is a **difference of squares** and can be further factored:
   \[
   (3x - 2)(x - 2)(x + 2)
   \]
   
So, the factored form is:
\[
(3x - 2)(x - 2)(x + 2)
\]

---

### b) \( x^2 + 11x + 30 \)

This is a quadratic trinomial. To factor it:
1. Find two numbers that multiply to 30 and add up to 11. These numbers are 5 and 6.
2. Factor the expression as:
   \[
   (x + 5)(x + 6)
   \]

So, the factored form is:
\[
(x + 5)(x + 6)
\]

---

### c) \( 6x^2 + 4xy + 3x^2y + 2y^2 \)

This is a quartic polynomial in two variables. Let's attempt to factor it by grouping:
1. Group the terms:
   \[
   (6x^2 + 4xy) + (3x^2y + 2y^2)
   \]
2. Factor out the common terms from each group:
   \[
   2x(3x + 2y) + y(3x + 2y)
   \]
3. Now, factor out the common binomial \( (3x + 2y) \):
   \[
   (3x + 2y)(2x + y)
   \]

So, the factored form is:
\[
(3x + 2y)(2x + y)
\]

---

Do you have any questions or need further clarification?

### Related Questions:
1. How do you apply the difference of squares rule?
2. What is the general strategy for factoring cubic polynomials?
3. Can every quadratic trinomial be factored? Why or why not?
4. How can we check that our factorization of a polynomial is correct?
5. What other methods can be used to factor higher degree polynomials?

### Tip:
When factoring by grouping, always look for common factors in pairs of terms and try to factor out common binomials.

Factor the following polynomials: a) 3x^3 - 2x^2 - 12x + 8, b) x^2 + 11x + 30, c) 6x^2 + 4xy + 3x^2y + 2y^2

Explore how to factor various polynomials, including cubic and quadratic forms, through detailed explanations and examples. Understand techniques like grouping, difference of squares, and handling variables in polynomials. Perfect for high school students mastering algebra.

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