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A nanny provides childcare and is a contributing member of a household, not a business. Business owners may directly deduct wages, payroll taxes, and certain benefits provided for the businesses' employees from their business income. Families do not have the ability to do the same on their personal income tax return. A family may only use the "Child and Dependant Care Tax Credit" or the "Dependent Care Flexible Spending Account" to use nanny expenses to reduce their income taxes.
So if your place of work doesn't off the flex account, you should go with the first option.
I have provided a link below on each topic for your reference from the IRS.
I hope this clarifies your question. Regards,