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A household employee can put the income Line 7 of Form 1040 and note “HSH” *ONLY* if the income is less the $1,800 (read the 1040 line 7 rules) .  The posters income was $4,500.   If the poster was actually a household employee and earned more than $1,800 then a W-2 is required from the employer, otherwise the poster is probably self-employed and must file a schedule C and SE to pay the employment tax.

Whether the babysitter is an employee or self-employed depends on the terms of the employment agreement.
**Disclaimer: This post is for discussion purposes only and is NOT tax advice. The author takes no responsibility for the accuracy of any information in this post.**