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Let's work through each part of the problem and factor the given expressions.

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

To factor this expression, we first find the greatest common factor (GCF) of the terms:
- The GCF of \( 8x^2 \) and \( 12x \) is \( 4x \).

Factoring out \( 4x \):
\[
8x^2 - 12x = 4x(2x - 3)
\]

### b) \( 9x^2 - 100 \)

This is a difference of squares. It can be factored using the formula \( a^2 - b^2 = (a - b)(a + b) \).
- Here, \( 9x^2 = (3x)^2 \) and \( 100 = 10^2 \).

Thus:
\[
9x^2 - 100 = (3x - 10)(3x + 10)
\]

### c) \( x^2 - 9x + 20 \)

We need to factor this quadratic. Look for two numbers that multiply to \( 20 \) and add up to \( -9 \).
- The numbers \( -4 \) and \( -5 \) satisfy these conditions.

Thus:
\[
x^2 - 9x + 20 = (x - 4)(x - 5)
\]

### d) \( 3x^2 - 4x - 7 \)

To factor this, we apply the method of factoring quadratics (splitting the middle term). We look for two numbers that multiply to \( 3 \times (-7) = -21 \) and add to \( -4 \).
- The numbers \( -7 \) and \( 3 \) satisfy these conditions.

We split the middle term and factor:
\[
3x^2 - 4x - 7 = 3x^2 - 7x + 3x - 7
\]
\[
= (3x + 7)(x - 1)
\]

### e) \( 14x^3 y + 21x^2 y^2 \)

The GCF of \( 14x^3 y \) and \( 21x^2 y^2 \) is \( 7x^2 y \).

Factoring out \( 7x^2 y \):
\[
14x^3 y + 21x^2 y^2 = 7x^2 y(2x + 3y)
\]

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Would you like further details on any of these steps?

Here are 5 relative questions to deepen your understanding:
1. How do you determine the greatest common factor in polynomial expressions?
2. Why does the difference of squares formula work, and when can you apply it?
3. Can you factor a quadratic equation using the quadratic formula instead of factoring by grouping?
4. What methods can be used for factoring more complicated trinomials?
5. How can you verify the factored form of a polynomial is correct?

**Tip**: Always double-check your factoring by expanding the factored form to ensure it matches the original expression.

Factorise each expression:
 a) 8x^2 - 12x
 b) 9x^2 - 100
 c) x^2 - 9x + 20
 d) 3x^2 - 4x - 7
 e) 14x^3y + 21x^2y^2

Learn how to factor various algebraic expressions such as 8x^2 - 12x and 9x^2 - 100 using methods like GCF and difference of squares. Detailed steps provided for Grades 9-10 students.

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