$$x = x^4-5x^2+4$$
Answer
$$ \begin{matrix}x_1 = 0.85844 & x_2 = -1.21831 & x_3 = -1.784 \\[1 em] x_4 = 2.14386 & \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x &= x^4-5x^2+4&& \text{move all terms to the left hand side } \\[1 em]x-x^4+5x^2-4 &= 0&& \text{simplify left side} \\[1 em]-x^4+5x^2+x-4 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using quartic formulas
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Polynomial Equations Solver