<s>[INST] Find the roots of the polynomial $x^4+2x^3-x-1=0$. Problem
Find all $4$ roots of the polynomial: $$f(x)=x^4+2x^3-x-1.$$

My Attempt
Observe,
$$f(-2)=1,\; f(-1)=-1,\; f(0)=-1,\; f(1)=1.$$
Therefore, $f(x)$ has a root between $-2$ and $-1$, another root between $0$ and $1$.
Since $f(x)$ does not contain a second degree term, a clever substitution might change $f(x)$ into a polynomial that is easier to deal with. For example $(x\to x-1)$,
$$f(x-1)=x^4-2x^3+x-1=f(-x)$$
Also, I am avoiding computational or graphical assistance.
 [/INST] With the substitution $x=t-b/na$, you eliminate the second highest order term of a polynomial $ax^n + bx^{n-1} + \cdots .$
If you substitute $x=t-2/4 = t-1/2$ your polynomial becomes
$$t^4 -\frac{3}{2}t^2 -\frac{11}{16}.$$
This is quadratic in $t^2$ so you can solve to get
$$t^2 = \frac{3\pm 2\sqrt{5}}{4}.$$
So the four roots of your polynomial are plus and minus the square roots of those two solutions (plus $1/2$ because of the original substitution.)  We were lucky that the first order term also disappeared.
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