Factorise the denominators first:
#(x+2y)/((x+4y)(x+4y)) + (x-4y)/((x+4y)(x+2y))#
#=color(white)(xxxxxxxxxxxxxxxxxxxx)/((x+4y)(x+4y)(x+2y))" "larr#find the LCD
#=(color(blue)((x+2y)(x+2y))+color(red)((x-4y)(x+4y)))/((x+4y)(x+4y)(x+2y))" "larr# see below
#=(color(blue)(x^2+4xy+4y^2) + color(red)(x^2 -16y^2))/((x+4y)(x+4y)(x+2y))" "larr# simplify
#=(2x^2+4xy-12y^2)/((x+4y)(x+4y)(x+2y))#
#=(2(x^2+2xy-6y^2))/((x+4y)(x+4y)(x+2y))#
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#(x+2y)/((x+4y)(x+4y)) xx (x+2y)/(x+2y) = ((x+2y)(x+2y))/((x+4y)(x+4y)(x+2y))#
#(x-4y)/((x+4y)(x+2y)) xx (x+4y)/(x+4y)= ((x+4y)(x-4y))/((x+4y)(x+4y)(x+2y))#
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#(a+b)^2 = a^2 +2ab +b^2" "larr# squaring a binomial
#color(blue)((x+2y)(x+2y) = x^2 +4xy+4y^2)#
#(a+b)(a-b) = a^2 -b^2" "larr# difference of two squares
#color(red)((x+4y)(x+4y)= x^2 -16y^2)#
