To solve this equation, we can expand the expressions inside the parentheses and combine like terms:

[(x + 1)^4] + [(x - 4)^4] = 97
(x^4 + 4x^3 + 6x^2 + 4x + 1) + (x^4 - 8x^3 + 16x^2 - 64x + 256) = 97
2x^4 - 4x^3 + 22x^2 - 60x + 257 = 97

Now, we have a fourth degree polynomial equation. We can try to simplify this equation further by moving all terms to one side:

2x^4 - 4x^3 + 22x^2 - 60x + 257 - 97 = 0
2x^4 - 4x^3 + 22x^2 - 60x + 160 = 0

Unfortunately, this polynomial cannot be easily factored. Therefore, a numerical or approximation method such as the Newton-Raphson method or graphing the equation would be needed to find the roots of this polynomial equation.