$$ \begin{aligned} z(z^4+1) &= 4+z^5&& \text{simplify left and right hand side} \\[1 em]z^5+z &= z^5+4&& \text{move all terms to the left hand side } \\[1 em]z^5+z-z^5-4 &= 0&& \text{simplify left side} \\[1 em]z^5+z-z^5-4 &= 0&& \\[1 em]z-4 &= 0&& \text{ move the constants to the right } \\[1 em]z &= 4&& \\[1 em]z &= 4&& \\[1 em] \end{aligned} $$

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