Strange things afoot
What is strange, there are four MB part numbers used in the HP Elitebook 8570 W, two that will accommodate a quad core CPU and two that will accommodate only a dual core, the quad core boards are supposed to have four memory slots while the dual core boards only two memory slots, your CPU is a quad core and so by HPs own tech sheet you should have a further two memory slots located beneath the keypad - next to the cooling fan, you have checked and Speccy supports what you have found, there are only the two memory slots on your MB, those that are located beneath the service door on the base of the notebook.
In case you are not already aware, when the HP Elitebook 8570 W version that has four memory slots shipped with only two of the slots populated the RAM was inserted in the two slots beneath the keypad, this then called the primary RAM, the two vacant slots beneath the service door were left empty for end user convenience and when fitted called expansion RAM
Your CPU is confirmed as an Intel Core i7 3720QM
The memory slots detected are;
Total memory slots: 2
Used memory slots: 2
Free memory slots: 0
If you take a look at the info below that I have highlighted in blue and red you may think like I am atm, either HP have used yet another version of the board in the HP Elitebook 8570 W or their parts list information is inaccurate.
System board (includes replacement thermal material)
For use with dual-core processors in all countries except China and Russia 690642-001
For use with quad-core processors in all countries except China and Russia 690643-001
For use with dual-core processors in China and Russia only 696440-001
For use with quad-core processors in China and Russia only 696441-001
NOTE:
A computer having a system board with spare part number
690643-001 or 696441-001 has four memory slots. In this case, the primary memory is located under the keyboard and expansion memory is located under the service door.
A computer having a system board with spare part number 690642-001 or 696440-001 has two memory module slots. In this case, the primary memory is located under the service door.
See HP info
here to find the exact details for your MB can you take a look and let us know which board it says you have.