An algebra problem by Sal Gard

Here was the method I used. Use synthetic division with -1 twice to get $x^4-2x^3-x^2+2x+1=0$ . Then divide by $x^2$ , and regroup the terms to get $(x^2-2+\frac{1}{x^2})+(-2x+\frac{2}{x})+1=0\Rightarrow (x-\frac{1}{x})^2-2(x-\frac{1}{x})+1=0\Rightarrow (x-\frac{1}{x}-1)^2=0 \Rightarrow x^2-x-1=0$ . The larger root of this equation is $\frac{\sqrt{5}+1}{2}$ .

Notice this should be the square of a function in order to be solved by immediate means. This will be proved later. By the rational root theorem, we can see $-1$ is a root. Now consider a number such that $x^2-x=1$ . By manipulation, we can see the roots of this and -1 would be added to a cubic that would be the square root. This can be done $x^2(x^2)-x^2(x)=x^2, x^2(x+1)-x(x+1)=x^2, x^3-x=x+1$ or $x^3-2x-1=0$ . If we square this, we get the desired expression. Now we see of the roots of $x^2-x=1$ , the golden ratio ( $(sqrt 5$ +1)/2) is the largest. So the answer is $101$ .