1. The problem is to find the value of $a - b + c$ given a parabola with vertex at $(-1, -4)$ and equation $f(x) = ax^2 + bx + c$. 2. The vertex formula for a parabola is $x_0 = -\frac{b}{2a}$ and $y_0 = c - \frac{b^2}{4a}$. 3. Given vertex coordinates are $x_0 = -1$ and $y_0 = -4$. 4. From $x_0 = -\frac{b}{2a} = -1$, we get $b = 2a$. 5. From $y_0 = c - \frac{b^2}{4a} = -4$, substitute $b = 2a$ to get $c - \frac{(2a)^2}{4a} = c - a = -4$ which implies $c = a - 4$. 6. Calculate $a - b + c = a - (2a) + (a - 4) = 0 - 4 = -4$. 7. Therefore, the final answer is $-4$.