1. **Problem statement:** We have a rectangular lawn with width $3x$ metres and area $15x^2 + 45x$ square metres. We want to find: a) An expression for the perimeter of the lawn. b) The number of fence posts needed if placed every 2 metres. 2. **Step a) Find the length:** The area $A$ of a rectangle is given by $A = \text{length} \times \text{width}$. We know width $w = 3x$ and area $A = 15x^2 + 45x$. 3. **Find length $l$:** $$l = \frac{A}{w} = \frac{15x^2 + 45x}{3x}$$ 4. **Simplify the length expression:** $$l = \frac{\cancel{3}5x^2 + \cancel{3}15x}{\cancel{3}x} = 5x + 15$$ 5. **Step a) Find perimeter $P$:** Perimeter of rectangle is $P = 2(l + w)$. Substitute $l = 5x + 15$ and $w = 3x$: $$P = 2((5x + 15) + 3x) = 2(8x + 15) = 16x + 30$$ 6. **Step b) Number of fence posts:** Fence posts every 2 metres means number of posts = $\frac{P}{2}$ plus 1 (to include both ends). $$\text{posts} = \frac{16x + 30}{2} + 1$$ 7. **Simplify posts expression:** $$\text{posts} = 8x + 15 + 1 = 8x + 16$$ **Final answers:** - Perimeter: $16x + 30$ metres - Number of fence posts: $8x + 16$ posts