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
