α & β are the roots of eqn. x²-p(1+x)-c =0

find value of ( α²+2α+1/α²+2α+c ) + (β²+2β+1/β²+2β+c )

1) 1

2) 2

3) 3

4) None of these

Dear Mayank

x²-p(1+x)-c =0

α & β are the roots of eqn

α²-p(1+α)-c =0   or α² =p(1+α) +c 

β²-p(1+β)-c =0   or β² =p(1+β) +c

and  α + β = p

     α *β =-p-c

NOw    ( α²+2α+1/α²+2α+c ) +  (β²+2β+1/β²+2β+c )

or       1 + (c-1)/(α²+2α+c )   + 1 + (c-1)/(β²+2β+c )

or       2  +(c-1)[1/(α²+2α+c )   +1/(β²+2β+c )]

or       2 + (c-1)[(α²+2α+c )   +(β²+2β+c ) ]/(α²+2α+c ) (β²+2β+c )

or       2 + (c-1) [α²+ β²+2(α+β)+2c]/[α²β² +2αβ(α+β) +c(α²+β²)+4αβ+2c(α+β) +c²]

 put the value

         2 + -1

      =1

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