1. **State the problem:** We are given the function $$f(x) = (x^2 - 6x)(x - 2) + 3x$$ and told it can be expressed as $$x(x^2 - 8x + 15)$$. We need to factorise $$f(x)$$ completely. 2. **Rewrite the function:** From the problem, we have $$f(x) = x(x^2 - 8x + 15)$$ 3. **Factor the quadratic:** Factorise the quadratic expression inside the parentheses: $$x^2 - 8x + 15$$ We look for two numbers that multiply to 15 and add to -8. These are -3 and -5. 4. **Write the factorised form:** $$x^2 - 8x + 15 = (x - 3)(x - 5)$$ 5. **Complete factorisation:** Substitute back: $$f(x) = x(x - 3)(x - 5)$$ **Final answer:** $$f(x) = x(x - 3)(x - 5)$$