\(x^4=4x+1\Leftrightarrow x^4-4x-1=0\)

\(\Leftrightarrow x^4+2x^2+1-2x^2-4x-2=0\)

\(\Leftrightarrow\left(x^2+1\right)^2-2\left(x+1\right)^2=0\)

\(\Leftrightarrow\left(x^2+1-\sqrt{2}x-\sqrt{2}\right)\left(x^2+1+\sqrt{2}x+\sqrt{2}\right)=0\)

1) \(x^2+1-\sqrt{2}x-\sqrt{2}=0\)

\(\Rightarrow x=\dfrac{\sqrt{2}\pm\sqrt{2-4\left(1-\sqrt{2}\right)}}{2}=\dfrac{\sqrt{2}\pm\sqrt{4\sqrt{2}-2}}{2}\)

2) \(x^2+1+\sqrt{2}x+\sqrt{2}=0\)

The equation has no roots since \(\left(\sqrt{2}\right)^2-4.1.\left(1+\sqrt{2}\right)< 0\)

So, the set of roots of the equation is \(S=\left\{\dfrac{\sqrt{2}\pm\sqrt{4\sqrt{2}-2}}{2}\right\}\)

x⁴ = 4x + 1

=> x⁴ - 4x - 1 = 0

=> \(\left(x^2\right)^2+2x^2+1-2x^2-4x-2=0\)

\(\Rightarrow\left(x^2+1\right)^2-2\left(x^2+2x+1\right)=0\)

\(\Rightarrow\left(x^2+1\right)^2-2\left(x+1\right)^2=0\)

\(\Rightarrow\left(x^2+1\right)^2=2\left(x+1\right)^2\)

\(\Rightarrow\sqrt{\left(x^2+1\right)^2}=\sqrt{2\left(x+1\right)^2}\)

\(\Rightarrow x^2+1=\sqrt{2}\cdot\left(x+1\right)\)

=> x = 1,663251939