1. **Problem:** Express the polynomial $x^2 + 6x + 12$ in terms of the functions $f(x) = x + 3$ and $g(x) = x^2$. 2. **Recall the functions:** - $f(x) = x + 3$ - $g(x) = x^2$ 3. **Rewrite the polynomial:** We notice that $x^2 + 6x + 12$ can be related to $f(x)$ and $g(x)$ by expanding $f(x)^2$: $$f(x)^2 = (x + 3)^2 = x^2 + 6x + 9$$ 4. **Compare with the given polynomial:** $$x^2 + 6x + 12 = f(x)^2 + 3$$ 5. **Answer for (i):** $$x^2 + 6x + 12 = g(x) + 6x + 12 = f(x)^2 + 3$$ --- 1. **Problem:** Express the polynomial $x^2 - 6x + 9$ in terms of $f(x)$ and $g(x)$. 2. **Recall:** $$f(x) = x + 3$$ 3. **Rewrite:** $$x^2 - 6x + 9 = (x - 3)^2$$ 4. **Express in terms of $f(x)$:** $$f(x) = x + 3 \\ herefore x - 3 = f(x) - 6$$ 5. **Square:** $$(x - 3)^2 = (f(x) - 6)^2 = f(x)^2 - 12f(x) + 36$$ 6. **Answer for (ii):** $$x^2 - 6x + 9 = (f(x) - 6)^2 = f(x)^2 - 12f(x) + 36$$