Electrical – add another cable to a box with four cables in it

What size box is it? It should either have a volume printed on it, or will be a standard size. Without knowing the box volume, we can't give you a definitive answer.

Assuming it's all 12 AWG, it's at least (12 (current carrying conductors) + 1 (grounds) + 4 (device fill)) * 2.25 = 38.25 cu. in. currently.

Adding another two conductor cable with ground would require an additional 4.5 cu. in. For a total of 42.75 cu. in., not counting any clamp fill not mentioned.

As long as the box is larger than this, you'll be okay.

Note that if you put the third device in, you'll have to add another 4.5 cu. in.

See this answer for more detail on box fill calculations.

You should definitely pull a new cable for the washing machine circuit.

Problems with your solution

• You can't use a red wire as a neutral (See NEC 200.6(A)).
• Circuit conductors (wires) must be contained in the same "raceway, auxiliary gutter, cable tray, cablebus assembly, trench, cable, or cord" (See NEC 300.3(B)).
• You have not clearly described where all the wires in the junction box come from/go to, or what else is on those circuits. So there's no way to know for sure if it's possible to use them to power the washing machine.
• Laundry branch circuits are not allowed to have any other outlets (See NEC 210.11(C)(2)).
• Laundry outlets must be GFCI protected (See NEC 210.8(A)(10)).