Problem - 120
Let $f(x) = x^4 + ax^3 + bx^2 + cx + d$. If $f(-1) = -1$, $f(2)=-4$, $f(-3) = -9$, and $f(4) = -16$. Find $f(1)$.
The answer is 23.
It is clear that -1, 2, -3, 4 are the four zeros of the polynomial $f(x) + x^2$.
Next, let $f(x) + x^2 = k(x+1)(x-2)(x+3)(x-4)$. Comparing the coefficient of the $x^4$ term leads to $k=1$.
Therefore $f(x)= (x+1)(x-2)(x+3)(x-4) - x^2$. Hence $f(1) = \boxed{23}$.