The first step is to multiply the top and bottom of each fraction by the denominator of the other fraction (sorry, this will be long!):
First Term:
#25/(x+sqrt(x^2+25))xx(-x+sqrt(x^2+25))/(-x+sqrt(x^2+25))#
#(25(-x+sqrt(x^2+25)))/((x+sqrt(x^2+25))(-x+sqrt(x^2+25)))#
#(25(-x+sqrt(x^2+25)))/(-x^2+(sqrt(x^2+25)^2)#
#(25(-x+sqrt(x^2+25)))/(cancel(-x^2+x^2)+25)#
#(cancel(25)(-x+sqrt(x^2+25)))/cancel(25)=color(red)(-x+sqrt(x^2+25))#
Second Term:
#-25/(-x+sqrt(x^2+25))xx(x+sqrt(x^2+25))/(x+sqrt(x^2+25))#
#-(25(x+sqrt(x^2+25)))/((-x+sqrt(x^2+25))(x+sqrt(x^2+25)))#
#-(25(x+sqrt(x^2+25)))/(-x^2+(sqrt(x^2+25)^2)#
#-(25(x+sqrt(x^2+25)))/(cancel(-x^2+x^2)+25)#
#-(cancel(25)(x+sqrt(x^2+25)))/cancel(25)=color(blue)(-(x+sqrt(x^2+25)))#
Put the first and second terms back together:
#color(red)(-x+sqrt(x^2+25))color(blue)(-(x+sqrt(x^2+25)))#
#color(red)(-x+cancel(sqrt(x^2+25)))color(blue)(-x-cancel(sqrt(x^2+25)))#
#color(red)(-x)color(blue)(-x)=color(green)(-2x)#