Renting to section 8

[Jane Dang]

I am listing my 5 bedrooms house for rent for $4200/month. A family likes the house and has a section 8 housing voucher for 4 bedrooms. Housing only pays $3641 for 4 bedrooms voucher. The family is willing to pay the difference out of their pocket. I want to know if it is legal to do so? Is it okay for me to rent 4bedroom voucher and collect the extra from the tenants? If so do I need to notify the housing office? Does anyone have experience on this case, please help me?