1. **State the problem:** We need to find the value of the function $f(x) = 3x^3 + 2x^2 + 1$ at $x=5$. 2. **Formula used:** The function is given by $$f(x) = 3x^3 + 2x^2 + 1$$ To find $f(5)$, substitute $x=5$ into the function. 3. **Substitute and calculate:** $$f(5) = 3(5)^3 + 2(5)^2 + 1$$ Calculate powers: $$= 3 \times 125 + 2 \times 25 + 1$$ 4. **Multiply:** $$= 375 + 50 + 1$$ 5. **Add all terms:** $$= 426$$ 6. **Final answer:** $$f(5) = 426$$