Solution of polynomial Find the set of values of $k$ for which the equation $x^4+kx^3+11x^2+kx+1=0$ has four distinct positive root.
Attempt: 
$x^4+kx^3+11x^2+kx+1=0$
$x^2+kx+11+{k\over x}+{1\over x^2}=0$
$x^2 + {1\over x^2} +k(x+{1\over x})+11=0$
$(x + {1\over x})^2 +k(x+{1\over x})+13=0$
I don't know how to proceed after this......

By your work let $f(u)=u^2+ku+13,$ where $u=x+\frac{1}{x}.$
Hence, $|u|>2$ and we need to solve the following system:
$$f(2)>0,$$
$$\frac{-k}{2}>2$$ and
$$k^2-52>0$$ or
$$f(-2)>0,$$
$$\frac{-k}{2}<-2$$ and
$$k^2-52>0,$$ which gives
$$-8.5<k<-\sqrt{52}$$ or
$$\sqrt{52}<k<8.5.$$
