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The expression to fully factorise is \( r^2 + 9r + 18 \).

Let's factorise this quadratic expression.

### Step-by-Step Factorisation:
1. We have the quadratic expression \( r^2 + 9r + 18 \), which is of the form \( ax^2 + bx + c \).
2. Here, \( a = 1 \), \( b = 9 \), and \( c = 18 \).
3. We need to find two numbers that multiply to \( c = 18 \) and add up to \( b = 9 \). The numbers are 3 and 6.
4. Therefore, we can rewrite the middle term:
   \[
   r^2 + 3r + 6r + 18
   \]
5. Now, factor by grouping:
   \[
   r(r + 3) + 6(r + 3)
   \]
6. Finally, factor out the common factor \( (r + 3) \):
   \[
   (r + 3)(r + 6)
   \]

So, the fully factorised form is:
\[
(r + 3)(r + 6)
\]

Let me know if you have any questions or if you'd like further details!

Here are 5 related questions:
1. How do you factor more complex quadratic expressions?
2. What method can you use when simple factorisation doesn't work?
3. How does the quadratic formula help in finding the roots of a quadratic equation?
4. What is the significance of factoring quadratic expressions?
5. Can all quadratic expressions be factored easily?

**Tip:** When factoring, always check if the product of the factors matches the original quadratic expression to verify correctness.

Learn how to fully factorise the quadratic expression r^2 + 9r + 18. This detailed solution involves factoring by grouping, a common technique in algebra. Ideal for students in Grades 8-10.

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