Let #f(x)=x^3+7x^2-x-7#
#f(1)=1+7-1-7=0#
so, #(x-1)# is a factor of #f(x)#
To find the other factors, we do a long division
#color(white)(aaaa)##x^3+7x^2-x-7##color(white)(aaaa)##|##color(blue)(x-1)#
#color(white)(aaaa)##x^3-x^2##color(white)(aaaaaaaaaaaa)##|##color(red)(x^2+8x+7)#
#color(white)(aaaaa)##0+8x^2-x#
#color(white)(aaaaaaa)##+8x^2-8x#
#color(white)(aaaaaaaaa)##+0+7x-7#
#color(white)(aaaaaaaaaaaaa)##+7x-7#
#color(white)(aaaaaaaaaaaaaa)##+0-0#
Therefore,
#f(x)=(x-1)(x^2+8x+7)=(x-1)(x+1)(x+7)#
Now, we can build the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-7##color(white)(aaaa)##-1##color(white)(aaaa)##1##color(white)(aaaaa)##+oo#
#color(white)(aaaa)##x+7##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x+1##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x+1##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##f(x)##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f(x)<0# when #x in ]-oo, -7 [uu]-1, 1[#
