**Category:** Calculus

**Subtype:** Function Analysis

**Problem Description:** The problem involves analyzing a continuous function \(f(x)\) defined on the interval \([-4, 4]\) and determining how many real values of \(m\) within this interval allow the function \(g(x) = |f(x^3 + 2x) + 3f(m)|\) to achieve a maximum value of 8 on the interval \([-1, 1]\).