The problem is how to say no without making them angry.
Literal
Problem [topic-は], how they [obj-を] not-angering [without-ずに] no say [Q-か] [copula-だ].
怒らせずに layers causative (怒らせる, 'make angry') with classical negative continuative ~ず + に ('without'). どのようにして is an elaborate 'by what means.' ノー is the English loanword 'no' used as a quoted word with いう.