Why does my laundry machine trip the GFCI when I plug it in?
So, your GFCI (Ground Fault Circuit Interrupter) trips every time you plug in your washer/dryer, but not when you plug in other equipment.

A GFCI detects ground faults, where the power supply line is shorted to ground; it then interrupts the circuit to protect you. Sounds like your washing machine has a ground fault, and your GFCI detects this and then trips. (This is a feature, not a bug.)

There's lots of things this could be: a failed motor, a wire with worn-through insulation, a circuit board failure, etc. etc. etc. Figuring out which would take a lot more information, but an initial test would be to measure the resistance between ground and the hot or neutral plug terminals. If you measure the resistance as anything other than infinite, you've confirmed your problem.