x4+4x3−8x2=−k ⇒f(x)=−k Where f(x)=x4+4x3−8x2=x2(x2+4x−8) Let g(x)=−k From the graph the following cases arise : 1. When −3≤−k≤0,⇒0≤k≤3 In this case, y=−k intersect at four points. 2. When −4≤−k<−3,⇒3<k≤4 In this case, y=−k intersect at two points, the given equation has two real roots. 3. When k<0,⇒−k>0 In this case, there are two points of intersection. So, the equation has two real roots.