Let's go through each factoring problem step by step:

1. Factor the expression $12x^3y^2 - 8x^2y^3 + 4xy$. Factor out the Greatest Common Factor (GCF).

• Step 1: Identify the GCF of the terms. The GCF of $12x^3y^2$, $8x^2y^3$, and $4xy$ is $4xy$.
• Step 2: Factor out $4xy$ from the expression: $12x^3y^2 - 8x^2y^3 + 4xy = 4xy(3x^2y - 2xy^2 + 1)$

2. Factor the polynomial $x^3 + 2x^2 - 2$. Factoring by Grouping

• Step 1: There are three terms, so let's try factoring by grouping, but here, we should start by attempting substitution or seeing if there's a special factorization:
• This polynomial does not factor nicely by grouping, and there's no simple factorization. Instead, we would have to use methods like the Rational Root Theorem, which is more advanced and indicates the polynomial does not have a simple factorization over the integers.

3. Factor $2x^2 + 7x + 3$. Factoring Quadratic Trinomials

• Step 1: Look for two numbers that multiply to $2 \times 3 = 6$ and add to $7$. These numbers are $6$ and $1$.
• Step 2: Rewrite the middle term using these numbers: $2x^2 + 6x + 1x + 3$
• Step 3: Factor by grouping: $(2x^2 + 6x) + (1x + 3) = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$

4. Factor $4x^2 + 12x + 9$. Factoring Perfect Square Trinomials

• Step 1: Recognize that this is a perfect square trinomial. The expression can be written as: $4x^2 + 12x + 9 = (2x + 3)^2$

5. Factor $25x^4 - 16y^2$. Factoring the Difference of Squares

• Step 1: Recognize this as a difference of squares: $25x^4 - 16y^2 = (5x^2)^2 - (4y)^2$
• Step 2: Factor using the difference of squares formula: $25x^4 - 16y^2 = (5x^2 - 4y)(5x^2 + 4y)$

6. Factor the cubic polynomial $x^3 - 3x^2 - 4x + 12$. Factoring Cubic Polynomials

• Step 1: Factor by grouping: $x^3 - 3x^2 - 4x + 12 = (x^3 - 3x^2) - (4x - 12)$
• Step 2: Factor out the common factors from each group: $x^2(x - 3) - 4(x - 3)$
• Step 3: Factor out the common binomial factor: $(x^2 - 4)(x - 3)$
• Step 4: Notice $x^2 - 4$ is a difference of squares: $(x + 2)(x - 2)(x - 3)$

Would you like more details or do you have any questions? Here are five related questions you might find helpful:

1. How can you factor polynomials with higher degrees like quartics?
2. What is the difference between a perfect square trinomial and a general trinomial?
3. How do you apply the Rational Root Theorem to factor cubic polynomials?
4. Can the method of factoring by grouping be applied to any polynomial?
5. How can you determine if a trinomial is a perfect square?

Tip: Always check for the greatest common factor first before proceeding with other factoring methods.