What is the ruling on a woman who vowed to recite four thousand glorifications (tasbih) daily, but she is unable to fulfill this due to her busy schedule? Is she sinful, and how can she expiate for her vow?

Undertaking a vow is disliked (makrooh), but it is obligatory to fulfill it if it is an act of obedience. You must make up for the missed vowed glorification (tasbeeh). If you are continuously unable to fulfill the vow, then its expiation is like the expiation for an oath: feeding ten poor people, or clothing them, or freeing a slave. If you are unable to do that, then fasting for three days.