Hey Paula, please consider renting a U-Haul Truck for transporting your mattress.
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"The 318600 can be used to secure furniture in U-Haul trucks. Other items such as the U-Haul 4-wheel dollies, furniture pads, Forearm Forklift and tie down rope are also very helpful when moving and securing furniture. For loading and driving tips, please see the U-Haul truck user instructions." http://www.uhaul.com/Articles/Tips/123/Truck-User-Instructions