To rationalize the denominator we need to multiply it by the appropriate form of #1#. For this form of denominator we use this rule of quadratics to determine what to multiply by:

#(color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y)) = color(red)(x)^2 - color(blue)(y)^2#

#(sqrt(color(red)(2)) - color(blue)(5))/(sqrt(color(red)(2)) - color(blue)(5)) xx (-3)/(sqrt(color(red)(2)) + color(blue)(5)) =>#

#(-3(sqrt(color(red)(2)) - color(blue)(5)))/(sqrt(color(red)(2))^2 - color(blue)(5)^2) =>#

#( (-3 xx sqrt(color(red)(2))) - (-3 xx color(blue)(5)))/(2 -25) =>#

#(-3sqrt(2) - (-15))/(2 -25) =>#

#(-3sqrt(2) + 15)/(-23) =>#

#-(15 - 3sqrt(2))/23#