A high school contest question Compute the largest root of $$x^4 −x^3 −5x^2 +2x+6$$
I cannot solved it by factor,what else I may do.
Such a question, how to think.

Note that $x^4-x^3-5x^2+2x+6=(x^4-5x^2+6)-x(x^2-2)=(x^2-2)(x^2-3)-x(x^2-2)=(x^2-2)(x^2-x-3)$. 
So we have two possible positive roots, $\sqrt{2}$ and $\frac{1+\sqrt{13}}{2}$. Notice that $\sqrt{13}>3$, so $\frac{1+\sqrt{13}}{2}>2>\sqrt{2}$. Thus, the largest root is $\frac{1+\sqrt{13}}{2}$.
