(a) In #P(x)=x^3+4x^2-x-4#, as the constant term is #-4#, the roots could be #{+-1,+-2,+-4}#.
(b) Further, sum of the coefficients is zero, one of the roots is #1#. As such #(x-1)# is a factor of #P(x)#. Dividing #P(x)# by #(x-1)# using synthetic division, we get
#1color(white)(X)|color(white)(X)1" "color(white)(X)4color(white)(XX)-1" "" "-4#
#color(white)(xx)|" "color(white)(XXx)1color(white)(XXX)5color(white)(XXXxX)4#
#" "stackrel("—————————————)#
#color(white)(xx)|color(white)(X)color(blue)1color(white)(X11)color(red)5color(white)(XXX)color(red)4color(white)(XXXxX)0#
(c) Hence #(x-1)# divides #P(x)# with quotient #x^2+5x+4# and as remainder is #0#, it is a factor. It is apparent that factors of #x^2+5x+4# are #(x+1)# and #(x+4)# i.e. #P(x)=(x-1)(x+1)(x+4)# and zeros are #-4,-1,1#.
(d) Hence, the graph of curve #P(x)=x^3+4x^2-x-4# cuts #x#-axis at #(-4,0),(-1,0),(1,0)#. We also have #P(-5)=-24#, #P(-3)=8#, #P(-2)=6#, #P(0)=-4#, #P(2)=18#.
(e) as #x->oo#, observe that #P(x)->oo# and as #x->-oo#, #P(x)->-oo#.
As differential of #P(x)# is #3x^2+8x-1# and this is zero at (using quadratic formula at #x=(-8+-sqrt(64+12))/2=(-4+-sqrt19)/3# i.e. #0.12# and #-2.79#, we have local extrema at these points.
As second derivative is #6x+8# and at #x=-2.79#, it is negative we have a local maxima at #x=-2.79# and second derivative at #x=0.12# is positive, we have a local minima at #x=0.12#.
(f) The graph appears as shown (not drawn to scale and shrank vertically to show the points discussed above) below:
graph{x^3+4x^2-x-4 [-6, 6, -30, 30]}
