\(Let $f(x) = 3x^2-2,$ and let $g(x)$ be a function such that $g(f(x)) = x^2 + x +1$. Find the sum of all possible values of $g(25)$. \)
+2666 We have \(25 = f(x) \).
Substituting what we know for \(f(x)\), we have the following equation: \(3x^2 - 2 = 25\)
Solving, we find \(x = \pm 3\).
If \(x = 3\), the value of \(g(f(x)) = 3^3 + 3 + 1 = 13\) and if \(x = -3\), \(g(f(x)) = (-3)^2 - 3 + 1 = 7\)
So, the sum of all possible values is \(13 + 7 = \color{brown}\boxed{20}\)
