Given #1/(x^2-3x+2)=1/(x+2)+5/(x^2-4)#
#(x^2-3x+2) = color(red)((x-2))color(blue)((x-1))#
#(x+2) = color(green)((x+2))#
#(x^2-4)= color(green)((x+2))color(red)((x-2))#
So the Least Common Denominator is
#color(white)("XXXX")##color(red)((x-2)color(green)((x+2))color(blue)((x-1))#
and #1/(x^2-3x+2)=1/(x+2)+5/(x^2-4)#
can be written as
#color(white)("XXXX")##(x+2)/(color(red)((x-2)color(green)((x+2))color(blue)((x-1)))) =( (x-2)(x-1) +5 (x-1))/(color(red)((x-2)color(green)((x+2))color(blue)((x-1)))#
If #x!=+-2# and #x!=1#
#color(white)("XXXX")##x+2 =( x^2-3x+2) +(5x-5)#
#color(white)("XXXX")##x+2 = x^2+2x-3#
#color(white)("XXXX")##x^2+x-3 = 0#
Using the quadratic formula to determine the roots
#color(white)("XXXX")##x=(-1+-sqrt(1^2-4(1)(-3)))/(2(1))#
#color(white)("XXXX")##x = (-1+-sqrt(13))/2#
