1. The problem is to find the smallest value of the expression $3n^2 - 10n - 14$ for integer values of $n$. 2. This is a quadratic expression in standard form $an^2 + bn + c$ with $a = 3$, $b = -10$, and $c = -14$. 3. To find the vertex (which gives the minimum value since $a > 0$), use the formula for the vertex $n = -\frac{b}{2a} = -\frac{-10}{2 \times 3} = \frac{10}{6} = \frac{5}{3} \approx 1.67$. 4. Since $n$ must be an integer, check values of $n = 1$ and $n = 2$. 5. Calculate $f(1) = 3(1)^2 - 10(1) - 14 = 3 - 10 - 14 = -21$. 6. Calculate $f(2) = 3(2)^2 - 10(2) - 14 = 12 - 20 - 14 = -22$. 7. Compare the two values: $f(2) = -22$ is less than $f(1) = -21$. 8. For completeness, also test $n=3$: $f(3) = 3(9) - 30 - 14 = 27 - 30 - 14 = -17$, which is higher. 9. Therefore, the smallest value of the expression for integer $n$ is $-22$, occurring at $n=2$. 10. Note: The mention of an ellipse centered at the origin with vertical major axis is not relevant to this expression; it seems to be extra information.