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I try to differentiate the equation, then I obtain 5x^4 + 1.

When at its local minimum or local maximum, 5x^4 + 1 = 0.

So, there is no solution for the equation, since 5x^4 + 1 > or = 1

So, I conclude that the graph of x^5 + x + c is increasing when x increases. So, I think the answer to this question is there is at most 1 root for the equation x^5 + x + c = 0. (Since c can take any value, we can shift up and down the graph, so there is at least one real root)

Is my answer correct? Or is there any other way to solve thi problem? Thanks