Mamy:
[tex]x^2+5x-1=0\\\\a=1,\ b=5,\ c=-1\\\\x_1+x_2=\dfrac{-b}{a}=\dfrac{-5}{1}=-5\\\\x_1 x_2=\dfrac{c}{a}=\dfrac{-1}{1}=-1[/tex]
Stąd:
[tex]x_1^4+x_2^4=(x_1^2+x_2^2)^2-2x_1^2x_2^2=[(x_1+x_2)^2-2x_1x_2]^2-2(x_1x_2)^2=\\\\=[(-5)^2-2\cdot(-1)]^2-2\cdot(-1)^2=[25+2]^2-2=27^2-2=729-2=\boxed{727}[/tex]
