2024 Mock AMC 12 - Problem 21

Let f\colon \mathbb{Z}^+ \to \mathbb{Z}^+ be a function satisfying f(x) = x^4 + 6x^2 + 2x - 5. Let S be the sum of all positive integers n such that f(n) is a perfect square. What is the sum of the digits of S^2?

\textbf{(A) }1\qquad\textbf{(B) }10\qquad\textbf{(C) }11\qquad\textbf{(D) }13\qquad\textbf{(E) }16