Prove that $x^7-2x^5+10x^2-1$ has no root greater than 1 (Second post)

Prove that $x^7-2x^5+10x^2-1$ has no root greater than 1.

In the previous blog post, I mentioned of solving the equation $x^7-2x^5+10x^2-1=0$ so to show that the function $x^7-2x^5+10x^2-1$ has no root greater than 1.

But, that is a really bad idea. The reason why the question setters stated the problem so mostly because they wanted to avoid us to solve for the problem. To solve for the polynomial of degree seven is really difficult, plus, the polynomial couldn't be factorized and so the real roots are kind of "ugly", there are no exact values for them.