find the condition that ` px^(3) + qx^(2) +rx +s=0` has exactly one real roots, where `p ,q ,r,s,in R`

Explanation

Let `f(x) = px^(3) + qx^(2) +rx +s`
so that ` f(x) = 3px^(2) +2px +r `
Now , ` f(x) =0` has all real roots, if f(x) =0 has two real roots
i.e. ` D ge 0`
` Rightarrow 4q^(2)-12pr ge 0`
` Rightarrow q^(2) -3pr ge 0`
But when ` q^(2) -3pr lt 0` , no root of f(x) =0 is real and hence f(x)=0 has exactly one real root