The Gateway uses SATA -150 interface while the HP Motherboard has both SATA 1 & 2 on it: HP and Compaq Desktop PCs - Motherboard Specifications, H-RS880-uATX (Aloe) | HP® Customer Support
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- One opt fan connector
- One cpu fan connector
- Four DDR3-DIMM sockets
- One 24-pin ATX power connector
- One front panel header
- Sata 1.0
- Sata 2.0
So you can install the Gateway Hard drive. But I would reformat the Hard drive to clean it of all data before installing it in the HP computer. The Windows installed doesn't have the drivers that the HP computer has for its hardware. Since you have a different CPU, you need to do a fresh Windows Installation. Plus the SATA 1 Hard drive will be very slow compared to the SATA 2 Hard drive installed. I would use the Gateway Hard drive as a backup in your computer.
This explains the difference between SATAs 1,2, & 3:
Difference between SATA I, SATA II and SATA III
What is the difference between SATA I, SATA II and SATA III?
SATA I (revision 1.x) interface, formally known as SATA 1.5Gb/s, is the first generation SATA interface running at 1.5 Gb/s. The bandwidth throughput, which is supported by the interface, is up to 150MB/s.
SATA II (revision 2.x) interface, formally known as SATA 3Gb/s, is a second generation SATA interface running at 3.0 Gb/s. The bandwidth throughput, which is supported by the interface, is up to 300MB/s.
SATA III (revision 3.x) interface, formally known as SATA 6Gb/s, is a third generation SATA interface running at 6.0Gb/s. The bandwidth throughput, which is supported by the interface, is up to 600MB/s. This interface is backwards compatible with SATA 3 Gb/s interface.
SATA II specifications provide backward compatibility to function on SATA I ports. SATA III specifications provide backward compatibility to function on SATA I and SATA II ports. However, the maximum speed of the drive will be slower due to the lower speed limitations of the port.
Example: SanDisk Extreme SSD, which supports SATA 6Gb/s interface and when connected to SATA 6Gb/s port, can reach up to 550/520MB/s sequential read and sequential write speed rates respectively. However, when the drive is connected to SATA 3 Gb/s port, it can reach up to 285/275MB/s sequential read and sequential write speed rates respectively.