If $|a|\ge 2$,$|b|\ge 2$, $P(x)=x^4-(a+b)x^3+(ab+2) x^2-(a+b)x+1$, prove that all roots of $P(x)$ are real 
Given: $a,b\in \mathbb R$, $|a|\ge 2$,$|b|\ge 2$,  $$P(x)=x^4-(a+b)x^3+(ab+2) x^2-(a+b)x+1,$$
Prove or disprove: all roots of $P(x)$ are real.

From a math contest. The polynomial is reciprocal, but I can't see how to use this on the proof, if that is the case, and I tried to use Descartes' rule of signs, but without success. I'm a little lost on how to address the problem.
Hints or solutions are appreciated. Sorry if this is a duplicate.

Hint...substitute $$u=x+\frac1x$$ after dividing thoughout by $x^2$
(This is the usual method with palindromic polynomials)
