1. **Problem:** Express the quadratic expressions in terms of the functions $f(x) = x + 3$ and $g(x) = x^2$. Specifically: (i) $x^2 + 6x + 12$ (ii) $x^2 - 6x + 9$ 2. **Recall the functions:** - $f(x) = x + 3$ - $g(x) = x^2$ 3. **Rewrite the expressions:** (i) $x^2 + 6x + 12$ Notice that $6x = 2 \times 3 \times x$ and $12 = 9 + 3$, so try to express in terms of $f(x)$ and $g(x)$. 4. **Express $f(x)^2$:** $$f(x)^2 = (x + 3)^2 = x^2 + 6x + 9$$ 5. **Compare with (i):** $$x^2 + 6x + 12 = f(x)^2 + 3$$ 6. **For (ii):** $$x^2 - 6x + 9 = (x - 3)^2$$ But $f(x) = x + 3$, so $x - 3 = f(x) - 6$ 7. **Express (ii) in terms of $f(x)$:** $$x^2 - 6x + 9 = (f(x) - 6)^2 = f(x)^2 - 12f(x) + 36$$ **Final answers:** - (i) $x^2 + 6x + 12 = f(x)^2 + 3$ - (ii) $x^2 - 6x + 9 = (f(x) - 6)^2 = f(x)^2 - 12f(x) + 36$