Question

5. If x + x-1 = a, express x2 + 2 + x-2 in terms of a.

Answer 1

I suppose the question is, if $x+x^{-1}=a$, what is $x^2+2+x^{-2}$?

Recall that $(a+b)^2=a^2+2ab+b^2$. It follows that for $x\neq0$,

$x^2+2+x^{-2}=x^2+2xx^{-1}+(x^{-1})^2=(x+x^{-1})^2=\boxed{a^2}$