If $f(x) = ax^2 + bx + c$ has an upward-opening parabola with roots between which $f(2) < 0$, and $f(-1) = 4(2k + 4) < 0$, find the range of $k$.
Step-by-Step Solution
Key Concept: If a point lies between the roots of an upward-opening parabola, the function value at that point is negative.
Since the parabola opens upward and $f(2) < 0$, the point $x = 2$ lies between the two roots. Also, $f(-1) = 4(2k + 4) < 0$ means $8k + 16 < 0$, so $k < -2$. Additionally, the constraint $4 + 2k + 4 - k - 3 < 0$ gives $k + 5 < 0$, hence $k < -5$.
Correct Answer: 1