1. **State the problem:** Factorise the quadratic expression $x^2 + 18x + 17$. 2. **Recall the formula:** To factorise a quadratic $ax^2 + bx + c$, we look for two numbers that multiply to $ac$ and add to $b$. 3. **Apply the formula:** Here, $a=1$, $b=18$, and $c=17$. We need two numbers that multiply to $1 \times 17 = 17$ and add to $18$. 4. **Find the numbers:** The factors of 17 are 1 and 17, and $1 + 17 = 18$, which matches $b$. 5. **Rewrite the middle term:** $$x^2 + 18x + 17 = x^2 + 1x + 17x + 17$$ 6. **Group terms:** $$= (x^2 + 1x) + (17x + 17)$$ 7. **Factor each group:** $$= x(x + 1) + 17(x + 1)$$ 8. **Factor out the common binomial:** $$= (x + 1)(x + 17)$$ **Final answer:** The factorised form is $$(x + 1)(x + 17)$$.