MHB Solving for Common Root in $(1)$ and $(2)$

[Albert1]

$(a-1)x^2-(a^2+2)x+(a^2+2a)=0----(1)\\
(b-1)x^2-(b^2+2)x+(b^2+2b)=0----(2)
$
if $(1)$ and $(2)$ have one root in common ,
(here $a,b\in N$ ,$a\neq b,\,\, and \,\, a>1,b>1$)
find value of :
$\dfrac{a^a+b^b}{a^{-b}+b^{-a}}$

 

[Albert1]

$(a-1)x^2-(a^2+2)x+(a^2+2a)=0----(1)\\
(b-1)x^2-(b^2+2)x+(b^2+2b)=0----(2)
$
if $(1)$ and $(2)$ have one root in common ,
(here $a,b\in N$ ,$a\neq b,\,\, and \,\, a>1,b>1$)
find value of :
$\dfrac{a^a+b^b}{a^{-b}+b^{-a}}$

hint

Spoiler

(1) and (2) can be factorized