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Let's solve each of the factorization problems step by step:

### a) \( x^2 + 6x + 8 \)

We look for two numbers that multiply to 8 (the constant term) and add to 6 (the coefficient of \(x\)).

- Factors of 8: \( (1, 8), (2, 4) \)
- \(2 + 4 = 6\), so the correct pair is 2 and 4.

Thus, we can factor this as:
\[
x^2 + 6x + 8 = (x + 2)(x + 4)
\]

### b) \( x^2 - 9x + 18 \)

We look for two numbers that multiply to 18 and add to -9.

- Factors of 18: \( (1, 18), (2, 9), (3, 6) \)
- \( -3 + -6 = -9 \), so the correct pair is -3 and -6.

Thus, the factorization is:
\[
x^2 - 9x + 18 = (x - 3)(x - 6)
\]

### c) \( x^2 - 8x - 9 \)

We need two numbers that multiply to -9 and add to -8.

- Factors of -9: \( (1, -9), (-1, 9), (3, -3) \)
- \(1 + (-9) = -8\), so the correct pair is 1 and -9.

Thus, the factorization is:
\[
x^2 - 8x - 9 = (x + 1)(x - 9)
\]

### d) \( x^2 + 6x - 27 \)

We look for two numbers that multiply to -27 and add to 6.

- Factors of -27: \( (1, -27), (-1, 27), (3, -9), (-3, 9) \)
- \(9 + (-3) = 6\), so the correct pair is 9 and -3.

Thus, the factorization is:
\[
x^2 + 6x - 27 = (x + 9)(x - 3)
\]

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If you have any further questions or need more details, feel free to ask!

Here are five follow-up questions to expand your understanding:
1. How would you factor \(x^2 + 7x + 10\)?
2. Can you explain how to complete the square for a quadratic equation?
3. What is the difference between factorization and solving a quadratic equation?
4. How can you determine whether a quadratic is factorable or not?
5. How do you solve a quadratic equation using the quadratic formula?

**Tip:** Always check your factors by expanding them back to the original expression. This ensures your factorization is correct.

Factorise the following: a) x^2 + 6x + 8 b) x^2 - 9x + 18 c) x^2 - 8x - 9 d) x^2 + 6x - 27

This guide explains how to factor quadratic equations such as x^2 + 6x + 8, x^2 - 9x + 18, x^2 - 8x - 9, and x^2 + 6x - 27. Learn the step-by-step process of factorizing and identifying key patterns in the equations. Ideal for students in Grades 8-10.

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