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Let's solve the given problems by factoring the provided polynomials.

### a. \( 3x^2 - 6x + 3 \)
We can factor out the greatest common factor (GCF) of all terms:
\[
3x^2 - 6x + 3 = 3(x^2 - 2x + 1)
\]
Now, factor the quadratic inside the parentheses:
\[
x^2 - 2x + 1 = (x - 1)(x - 1) = (x - 1)^2
\]
Thus, the factored form is:
\[
3(x - 1)^2
\]

### b. \( -4x^3 + 2x^2 + 6x \)
First, factor out the greatest common factor (GCF):
\[
-4x^3 + 2x^2 + 6x = -2x(2x^2 - x - 3)
\]
Now, we factor the quadratic \( 2x^2 - x - 3 \). Look for two numbers that multiply to \( 2(-3) = -6 \) and add to \( -1 \). These numbers are \( -3 \) and \( 2 \). Rewrite the middle term:
\[
2x^2 - 3x + 2x - 3 = (2x^2 - 3x) + (2x - 3)
\]
Factor by grouping:
\[
= x(2x - 3) + 1(2x - 3) = (2x - 3)(x + 1)
\]
Thus, the factored form is:
\[
-2x(2x - 3)(x + 1)
\]

### c. \( 15x^2 + 7x - 2 \)
We need to factor this quadratic. Find two numbers that multiply to \( 15(-2) = -30 \) and add to \( 7 \). These numbers are \( 10 \) and \( -3 \). Rewrite the middle term:
\[
15x^2 + 10x - 3x - 2 = (15x^2 + 10x) - (3x + 2)
\]
Factor by grouping:
\[
= 5x(3x + 2) - 1(3x + 2) = (3x + 2)(5x - 1)
\]
Thus, the factored form is:
\[
(3x + 2)(5x - 1)
\]

### d. \( x^3 - x + 2x^2 - 2 \)
First, rearrange the terms in standard form:
\[
x^3 + 2x^2 - x - 2
\]
Now, factor by grouping:
\[
(x^3 + 2x^2) - (x + 2) = x^2(x + 2) - 1(x + 2)
\]
Factor out the common binomial factor:
\[
= (x^2 - 1)(x + 2)
\]
Notice that \( x^2 - 1 \) is a difference of squares:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Thus, the factored form is:
\[
(x - 1)(x + 1)(x + 2)
\]

Would you like further details on any of these factorizations or have any questions?

Here are five related questions:
1. How do you factor a quadratic trinomial?
2. What are the steps for factoring by grouping?
3. How do you recognize a difference of squares?
4. Can you always factor a cubic polynomial?
5. What is the significance of finding the greatest common factor (GCF) first?

**Tip:** Always check for the GCF before trying more advanced factoring techniques. It simplifies the polynomial and makes the process easier.

Factor the following polynomials: a. 3x^2 - 6x + 3, b. -4x^3 + 2x^2 + 6x, c. 15x^2 + 7x - 2, d. x^3 - x + 2x^2 - 2

Learn how to factor different types of polynomials, including quadratic and cubic expressions. This guide covers factoring by grouping, using the greatest common factor, and recognizing the difference of squares. Suitable for high school students in Grades 9-12.

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