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Let's go ahead and factorize the given expressions step by step:

### (a) \( 14p - 21 \)
Factor out the greatest common factor (GCF):
\[
14p - 21 = 7(2p - 3)
\]

### (b) \( 9x - 12x^2 \)
Factor out the GCF:
\[
9x - 12x^2 = 3x(3 - 4x)
\]

### (c) \( 8c^2 - 24c \)
Factor out the GCF:
\[
8c^2 - 24c = 8c(c - 3)
\]

### (d) \( 30g + 54gh \)
Factor out the GCF:
\[
30g + 54gh = 6g(5 + 9h)
\]

### (e) \( a^3b - a^2b^2 - b^3 \)
This can be factored by grouping. First, rewrite the expression:
\[
a^3b - a^2b^2 - b^3 = ab(a^2 - ab) - b^3
\]
Now, factor by grouping:
\[
ab(a^2 - ab) - b^3 = b(a^3 - ab^2)
\]

### (f) \( 6p + 8p^2 - 4p^3 \)
Factor out the GCF:
\[
6p + 8p^2 - 4p^3 = 2p(3 + 4p - 2p^2)
\]

### (g) \( x^4y^2 - x^2y^4 - x^4y^4 \)
Factor out the GCF:
\[
x^4y^2 - x^2y^4 - x^4y^4 = x^2y^2(x^2 - y^2 - x^2y^2)
\]

### (h) \( 15y^2z^3 - 20y^3z^4 + 35y^2z^2 \)
Factor out the GCF:
\[
15y^2z^3 - 20y^3z^4 + 35y^2z^2 = 5y^2z^2(3z - 4yz^2 + 7)
\]

### (i) \( 10p^2q^2 - 2pq + 8q^3 \)
Factor out the GCF:
\[
10p^2q^2 - 2pq + 8q^3 = 2q(5p^2q - p + 4q^2)
\]

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Would you like further details on any of these steps, or do you have any specific questions?

Here are 5 related questions for further practice:

1. How do you determine the greatest common factor (GCF) of terms?
2. What is factorization by grouping, and how do you apply it?
3. How can we factor expressions involving more than one variable?
4. What is the role of negative signs in factorizing?
5. How can you factor trinomials with higher powers of variables?

**Tip:** Always look for common factors across all terms before trying more advanced factorization methods like grouping or the quadratic formula.

Factorise the following expressions: (a) 14p - 21, (b) 9x - 12x^2, (c) 8c^2 - 24c, (d) 30g + 54gh, (e) a^3b - a^2b^2 - b^3, (f) 6p + 8p^2 - 4p^3, (g) x^4y^2 - x^2y^4 - x^4y^4, (h) 15y^2z^3 - 20y^3z^4 + 35y^2z^2, (i) 10p^2q^2 - 2pq + 8q^3

Learn how to factorize various algebraic expressions with step-by-step solutions. This page covers polynomial factorization techniques including GCF extraction, factoring by grouping, and handling complex terms with multiple variables. Suitable for Grades 9-11 students.

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