x^3(x+1)=(x+k)(x+2k)where k belongs to (0.75,1)find-i)no. of real rootsii) greatest real rootiii) least real rootplz help me to solve this or guide me approaching this sum and more such types.......thanks in advance........

The no of roots of this equation are the no of points of intersection of y=x^4+x^3 and y=x^2+3kx+2k^2.

Differentiating y=x^4+x^3 wrt x we get dy/dx=4x^3+3x^2=x^2(4x+3). dy/dx>0 when x> -3/4, dy/dx

The graph of y=x^2+3kx+2k^2= (x-3k/2)^2-k^2/4 is a parabola with vertex at

(-3k/2,-k^2/4) By plotting these graphs we observe that there are two points of intersections. So the equation has two distinct real roots for each k belongs to (0.75,1) .

The no of roots of this equation are the no of points of intersection of y=x^4+x^3 and y=x^2+3kx+2k^2.
Differentiating y=x^4+x^3 wrt x we get dy/dx=4x^3+3x^2=x^2(4x+3). dy/dx>0 when x> -3/4, dy/dx
The graph of y=x^2+3kx+2k^2= (x-3k/2)^2-k^2/4 is a parabola with vertex at
(-3k/2,-k^2/4) By plotting these graphs we observe that there are two points of intersections. So the equation has two distinct real roots for each k belongs to (0.75,1) .