Let #f(x)=x^3+4x^2-x-4#
Then, #f(1)=1+4-1-4=0#
Therefore, #(x-1)# is a factor of #f(x)#
To find the other factors, let's do a long division
#color(white)(aaaaa)##x^3+4x^2-x-4##color(white)(aaaaa)##∣##x-1#
#color(white)(aaaaa)##x^3-x^2##color(white)(aaaaaaaaaaaaa)##∣##x^2+5x+4#
#color(white)(aaaaaa)##0+5x^2-x#
#color(white)(aaaaaaaa)##+5x^2-5x#
#color(white)(aaaaaaaaaaaa)##0+4x-4#
#color(white)(aaaaaaaaaaaaaa)##+4x-4#
#color(white)(aaaaaaaaaaaaaaa)##+0-0#
Therefore, #f(x)=(x-1)(x+1)(x+4)>=0#
Let's do the sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaa)##-1##color(white)(aaaa)##+1##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x+4##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x+1##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##x-1##color(white)(aaaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaaa)##f(x)##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore #f(x)>=0# if #x in [-4,-1] uu [1, +oo] #
graph{x^3+4x^2-x-4 [-20.28, 20.27, -10.13, 10.13]}