For negative values, x^(1/3) can have 3 different values. The default meaning of x^(1/3) is not to take the real value. To see this evaluateN[(-1)^(1/3)]. The default value is the first root counter clockwise from the real positive axis. There's a method a method and reason to this - it provides a consistent interpretation for what the nth root of a negative number returns.

There is a function which returns the Real value of the root. It's called Surd:

Plot[Surd[x, 3]^4 + 4*Surd[x, 3], {x, -5, 5}]