[tex]\displaystyle \sf x^4+x^3+x^2+x+1= 0 \ \div(x^2)\\\\ \frac{x^4}{x^2}+\frac{x^3}{x^2}+\frac{x^2}{x^2}+\frac{x}{x^2}+\frac{1}{x^2}=0 \\\\\\ x^2+x+1+\frac{1}{x}+\frac{1}{x^2}=0 \\\\\ \left(x^2+\frac{1}{x^2}\right) + \left(x+\frac{1}{x}\right)+1=0 \\\\\\\ \text{Fa\c ca }:\\\\ \left(x+\frac{1}{x}\right) = m \to \left(x^2+\frac{1}{x^2}\right)=m^2-2\\\\\ \\ Assim,\\\\ m^2-2+m+1 = 0 \\\\ m}^2+m-1=0 \\\\ m=\frac{-1\pm\sqrt{1^2-4\cdot (-1).1}}{2.1}[/tex]

[tex]\displaystyle \sf m = \frac{-1\pm\sqrt{5}}{2} \\\\\\ Temos : \\\\\ \left(x+\frac{1}{x}\right) = \frac{-1+\sqrt{5}}{2}\ \ ;\ \ \left(x+\frac{1}{x}\right) = \frac{-1-\sqrt{5}}{2} \\\\\\ 1\º ) \\\\ \left(x+\frac{1}{x}\right) = \frac{-1+\sqrt{5}}{2} \\\\ \\ 2x^2+2=x(-1+\sqrt{5}) \\\\2x^2-x(-1+\sqrt{5})+2=0 \\\\ x= \frac{-(-(-1+\sqrt{5}))\pm\sqrt{(-1+\sqrt{5})^2-4\cdot2\cdot 2}}{2\cdot 2}\\\\\\ x= \frac{\sqrt{5}-1\pm\sqrt{6-2\sqrt{5}-16 }}{4}[/tex]

[tex]\displaystyle \sf \boxed{\sf \ x=\frac{\sqrt{5}-1\pm\sqrt{-10-2\sqrt{5}}}{4} \ }\ { \not \in \mathbb{R} }[/tex]

[tex]\displaystyle \sf 2\º ) \\\\ \left(x+\frac{1}{x}\right) = \frac{-1-\sqrt{5}}{2} \\\\ \\ 2x^2+2=x(-1-\sqrt{5}) \\\\2x^2+x(1+\sqrt{5})+2=0 \\\\ x= \frac{-(1+\sqrt{5})\pm\sqrt{(1+\sqrt{5})^2-4\cdot2\cdot 2}}{2\cdot 2}\\\\\\ x= \frac{-\sqrt{5}-1\pm\sqrt{6+2\sqrt{5}-16 }}{4} \\\\\\ \boxed{\sf \ x= \frac{-\sqrt{5}-1\pm\sqrt{-10+2\sqrt{5} }}{4} \ }\ \not\in\mathbb{R}[/tex]

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