1. **Stating the problem:** We need to find constants $a$, $b$, and $c$ such that the function $$f(x) = (x + 3)(ax^2 + bx + c)$$ matches a given expression or condition (not explicitly provided in the question, so we assume we want to express $f(x)$ in standard polynomial form by expanding and comparing coefficients. 2. **Formula and approach:** To find $a$, $b$, and $c$, we expand the product: $$f(x) = (x + 3)(ax^2 + bx + c) = x(ax^2 + bx + c) + 3(ax^2 + bx + c)$$ 3. **Expanding:** $$f(x) = a x^3 + b x^2 + c x + 3 a x^2 + 3 b x + 3 c$$ 4. **Combine like terms:** $$f(x) = a x^3 + (b + 3 a) x^2 + (c + 3 b) x + 3 c$$ 5. **Interpretation:** The polynomial $f(x)$ is now expressed as $$f(x) = a x^3 + (b + 3 a) x^2 + (c + 3 b) x + 3 c$$ 6. **Next step:** To find $a$, $b$, and $c$, we need the explicit form of $f(x)$ or conditions to match coefficients. Since the problem does not provide that, this is the general form of $f(x)$ in terms of $a$, $b$, and $c$. If you provide the explicit polynomial $f(x)$ or conditions, we can solve for $a$, $b$, and $c$ by equating coefficients. **Final answer:** $$f(x) = a x^3 + (b + 3 a) x^2 + (c + 3 b) x + 3 c$$ This expresses $f(x)$ in terms of $a$, $b$, and $c$ after expansion.