Solve polynomial equation with real parameter

Question

Solve the equation $x^4-(2m+1)x^3+(m-1)x^2+(2m^2+1)x+m=0,$ where $m$ is a real parameter.

My work: So far I've been able to factor the polynomial to $(-x^2+x+m)(-x^2+2mx+1)=0$. Then after using the quadratic formula with each of the factors I'm here: $x=\frac{-1 \pm \sqrt{1+4m}}{-2}$ and $x=\frac{-2m \pm \sqrt{4m^2+4}}{-2}$

Answer 1

Take the negatives out. It will look tidier. So: $$(x^2-x-m)(x^2-2mx-1)=0$$ Then: $$x=\frac{1\pm\sqrt{1-4m}}{2}=\frac 12\pm\frac{\sqrt{1-4m}}{2}$$ And: $$x=\frac{2m\pm\sqrt{4m^2+4}}{2}=m\pm\sqrt{m^2+1}$$ You can't do anything more here.

Answer 2

It seems that you are not sure if they are all roots. The fundamental theorem of algebra implies that a polynomial of positive degree $n$ has exactly $n$ roots. In your case, there should be 4 roots and you have found them.