We know that,
" If , # l and m# are the roots of #Ax^2+Bx+C=0# ,then
#(i)# the SUM of the Roots #=l+m=-B/A#
#(ii)# the PRODUCT of the Roots #=l*m=C/A#
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Here,
#alpha and beta# are the roots of #x^2+ax-b=0#
So, #color(red)(alpha+beta=-a ,and alpha*beta=-bto(1)#
We have,
#gamma and delta # are the roots of #x^2+ax+b=0#
So, #color(red)(gamma+delta=-a , and gamma*delta=bto(2)#
From #(1) and (2)#
#color(blue)(gamma+delta=alpha+beta and gamma*delta=-alpha*betato(3)#
Let,
#K=(alpha-gamma)(alpha-delta)(beta-delta)(beta-gamma)#
#K=[alpha^2-alphagamma-alphadelta+gamma*delta][beta^2-gammabeta-deltabeta+gamma*delta]#
#K=[alpha^2-alphacolor(blue)((gamma+delta)+gamma*delta)][beta^2-betacolor(blue)((gamma+delta)+gamma*delta)]#
Using #(3)#we get
#K=[alpha^2-alphacolor(blue)((alpha+beta)-alpha*beta)][beta^2-betacolor(blue)((alpha+beta)-alphabeta)]#
#K=[alpha^2-alpha^2-alphabeta-alphabeta][beta^2-alphabeta-beta^2-alphabeta]#
#K=[-2alphabeta][-2alphabeta]#
#K=4(alphabeta)^2 tocolor(red)(Apply(1)#
#K=4(-b)^2=4b^2#
Hence,
#(alpha-gamma)(alpha-delta)(beta-delta)(beta-
gamma)=4(alphabeta)^2=4b^2#