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The task here is to find the common factor from each expression and then match it with the appropriate solution from the list on the right-hand side. Let's go step by step.

### a) \( 4x^2 - 6x + 2x^3 \)

We can rearrange terms to have the highest degree first:
\[
2x^3 + 4x^2 - 6x = 2x(x^2 + 2x - 3)
\]
This matches with the solution: **\( 2x(x^2 + 2x - 3) \)**.

### b) \( 3x^3 + 6x^2 - 12x \)

We can factor out the greatest common factor:
\[
3x(x^2 + 2x - 4)
\]
This matches with the solution: **\( 3x(x^2 + 2x - 4) \)**.

### c) \( 12x^4y^2 + 6x^2y^4 - 15x^3y \)

The common factor here is \( 3x^2y \):
\[
3x^2y(4x^2 + 2y^3 - 5x)
\]
This matches with the solution: **\( 3x^2y(4x^2 + 2y^3 - 5x) \)**.

### d) \( -12x^3 - 8x^4 + 4x^2 + 4x^6 \)

We can rearrange terms and factor:
\[
-4x^2(3x + 2x^2 - 1 - x^4)
\]
This matches with **\( 4x^2(x^4 - 2x^2 - 3x + 1) \)**.

### e) \( 8x^2 - 8x^3 \)

Factor out \( 8x^2 \):
\[
8x^2(1 - x)
\]
This matches with: **\( 8x^2(1 - x) \)**.

### f) \( -3xy - 2xy^2 - 10x^2yz \)

The common factor is \( -xy \):
\[
-xy(3 + 2y + 10xz)
\]
This matches with: **\( -xy(3 + 2y + 10xz) \)**.

### g) \( -3x + 6x^2 + 12x^3 \)

Factor out \( -3x \):
\[
-3x(-4x^2 - 2x + 1)
\]
This matches with: **\( 3x(4x^2 + 2x - 1) \)**.

### h) \( 2ab^2 - 4a^3b + 8a^4b^3 \)

Factor out the common term:
\[
2ab(b - 2a^2 + 4a^3b^2)
\]
This matches with: **\( 2ab(b - 2a^2 + 4a^3b^2) \)**.

### Final Matching:
a) \( 2x(x^2 + 2x - 3) \)  
b) \( 3x(x^2 + 2x - 4) \)  
c) \( 3x^2y(4x^2 + 2y^3 - 5x) \)  
d) \( 4x^2(x^4 - 2x^2 - 3x + 1) \)  
e) \( 8x^2(1 - x) \)  
f) \( -xy(3 + 2y + 10xz) \)  
g) \( 3x(4x^2 + 2x - 1) \)  
h) \( 2ab(b - 2a^2 + 4a^3b^2) \)

Would you like more details on any of these steps? Here are 5 related questions:

1. How can you identify the greatest common factor in an expression?
2. What are some strategies for factoring complex polynomials?
3. Why is it useful to factor expressions in algebra?
4. Can factoring always simplify an equation to its fullest extent?
5. How can you apply factoring in solving equations?

**Tip:** When factoring, always check for the greatest common factor first, as it simplifies the process significantly.

Extract common factor in each case and match with the correct expression.

Learn how to extract common factors from polynomial expressions and match them to their factored forms. This exercise covers key algebraic concepts suitable for students in Grades 9-11.

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