1. **State the problem:** We need to find the expanded form of the function $f(x) = (3x^2 + x)^2$. 2. **Formula used:** To expand a square of a binomial, use the formula $$(a + b)^2 = a^2 + 2ab + b^2$$ where $a = 3x^2$ and $b = x$. 3. **Apply the formula:** $$f(x) = (3x^2)^2 + 2 \cdot (3x^2) \cdot x + x^2$$ 4. **Calculate each term:** - $(3x^2)^2 = 9x^4$ - $2 \cdot (3x^2) \cdot x = 6x^3$ - $x^2 = x^2$ 5. **Combine all terms:** $$f(x) = 9x^4 + 6x^3 + x^2$$ 6. **Final answer:** $$\boxed{9x^4 + 6x^3 + x^2}$$