Beginner Exercise 269 / Solution

Solution  

is the solution. If connects, then connects the whole group back. (If at a, then White plays as well, or or b.)

Failure 1  

Black wins the capturing race, three liberties to two. If White resists with at a, then at b. After this, any can be met with capturing at c, and if takes c, then at d puts the white group in atari.

Failure 2  

here is a better defense, but Black can get a ko.