I have a Franklin Electric 208-240 V 1/4 HP 1050 rpm cont air over electric motor. Other numbers are 8745860160 and 51-20177-01-02. 60Hz 2.0 amp 5MFD/370VAC. It has 7 wires coming out of it. They are: two orange, 1 red, 1black, 1 yellow, 1 blue, 1 brown plus a green pig tail bolted to the case. I have a two terminal capacitor 7.5 uf 440 V 60 Hz also with a case grounding green pig tail.
I want to know which color wires I should hook up to the capacitor and which color wires I should connect to the two house mains of 120V.

This won't help you much but,

8745860160 Is the manufactures or model number
51-20177-01-02 is probably a manufactures date and serial number

5MFD/370VAC is the size capacitor required.

You have a 7.5 uf 440 V capacitor

Note that the voltage on a capacitor is the max voltage that can be applied. Not a required voltage.
You of course have 240 volts.

The micro farads is the storage capacity of a capacitor.
Motor requires 5MFD capacitor, you have 7.5MFD capacitor (7.5 uf means 7.5 MFD). Should be OK.

This is a 3 speed motor.

Look on the motor label. There you should find the lead identification or diagram.
The red and black are probably the high voltage input leads. The orange are probably the capacitor leads. The yellow, blue and brown are probably the speed control leads. But that is just a guess. Look for diagram on motor. Assuming I am correct that orange are capacitor leads, it matter not which lead connects to which terminal on the capacitor.

Depending on you application you may or may not use all 3 speeds. For instance you may use 1 for heat and 2 for cooling or 2 for heat and 3 for cooling. Takes more speed to move colder air.

Sorry but all the HVAC technicians left this site some time ago.