1. **State the problem:** Solve the quadratic expression $\frac{1}{4}x^2 + 3x + 9$ or simplify it if possible. 2. **Formula and rules:** This is a quadratic expression in standard form $ax^2 + bx + c$ where $a=\frac{1}{4}$, $b=3$, and $c=9$. 3. **Simplify or rewrite:** To work with it more easily, multiply the entire expression by 4 to clear the fraction: $$4 \times \left(\frac{1}{4}x^2 + 3x + 9\right) = 4 \times \frac{1}{4}x^2 + 4 \times 3x + 4 \times 9$$ $$= \cancel{4} \times \frac{1}{\cancel{4}} x^2 + 12x + 36$$ $$= x^2 + 12x + 36$$ 4. **Factor the quadratic:** Recognize that $x^2 + 12x + 36$ is a perfect square trinomial: $$x^2 + 12x + 36 = (x + 6)^2$$ 5. **Rewrite the original expression:** Since we multiplied by 4 earlier, divide back by 4: $$\frac{1}{4}x^2 + 3x + 9 = \frac{(x + 6)^2}{4} = \left(\frac{x + 6}{2}\right)^2$$ **Final answer:** $$\frac{1}{4}x^2 + 3x + 9 = \left(\frac{x + 6}{2}\right)^2$$