To solve this quartic equation, we can try to factor it or use numerical methods to find the roots.

One way to potentially factor the equation is by grouping:

X^4 - 8x^3 - 4x^2 + 16x + 4 = 0

Factor out common terms:

x^3(x - 8) - 4(x^2 - 4) + 4 = 0

(x^3 - 4)(x - 8) + 4 = 0

(x^3 - 4)(x - 8) = -4

(x^3 - 4) = -4 / (x - 8)

x^3 = -4 / (x - 8)

This factorization doesn't provide a clear way to further simplify the equation. Thus, we can use numerical methods such as the Newton-Raphson method or the Numerical Analysis method to approximate the roots of the equation, or use a computer algebra system to solve it.