Working backwards is IMO a great way to do it, and probably the most direct path. It is how I would do it for sure. I'd even argue it's hardly backwards when you are going from what they gave you (remaining peanuts) to the answer (starting peanuts). And it can totally be done formulaically once you work out what operation 'undoes' taking some fraction of the peanuts. That's pretty clear for taking 1/2 (multiply by 2), but what about taking 1/4? Well that means the peanuts left was 3/4 of what was there before. The amount of peanuts taken (1/4 of the original) is then 1/3 of the peanuts left (3/4 of the original). So you add on 1/3 of what you had, or in other words multiply by 4/3. You can also think about that by saying you got to what you have left by multiplying by the 3/4 remaining after taking 1/4, so to get what you started do the opposite and divide by 3/4 (equivalent to multiplying 4/3). Once you see how to do that it is pretty quick to write out the formula:

(71+10)\*2\*4/3\*3/2 = 324

That said the way the other commenter did it by first writing out the situation as described with a variable:

x\*(1-1/3)\*(1-1/4)\*(1-1/2)-10=71

and then using the consistent and established rules of algebra to get the answer from there is also good (if you do that you might see you end up doing exactly what I described in the first section anyway, add 10, and then divide by 1/2, 3/4, and 2/3). But doing it algebraically like this may leave potentially less room for error (and if there is an error, a more detectable one) as it clearly delineates the process of modelling the problem in the language of math and then solving the problem using a clearly defined mathematical toolset rather than sort of doing both as you go working backwards one operation at a time.

Ultimately there are many ways to go about it, and IMO one of the beauties of math is that as long as you know and understand what path you are taking, you'll arrive at the same answer as someone who might have seen and worked the problem with a completely different line of reasoning. The path you take to the answer is really much less important than building a good understanding and framing of the problem where you are confident in your reasoning toward that solution.