Not any great easy way I can think of to do that one but I would attempt to do 400 by 3974 and then add chunks of 438 x 10 or x5 until I got really close and then add individual blocks.

So like 400 by 3974, you can round to 4000 and remove 4 x 26 = 104 after doubling 4000 twice. So we have 4000 to 8000 to 16000 remove 104 is 15896, add zeros is 1,589,600. Forget all other numbers but this one.

We are missing 38 x 3974. We can do the same round and remove trick to add 10 x 3974 by changing it to 10 x 4000 - 10 x 26. We need four of those though, so we can double it and turn from 40000 - 260 to 80000 - 520 and then 160,000 - 1040 or 158,960. Need to remove 2 x 3974 though, so remove 8000 and add 52 so 151,012.

Hopefully ive been able to keep that first number fresh in my head this whole time, which involves repeating it for me, and I'd add 1,589,600 and 151,012. Add 150000 and then 1,012 so 1,739,600 and then 1,740,612.

That all said, I make way more mistakes than a calculator, and I was off by 400 or so on my first run through. Also its really easy to forget big numbers like that for me. I'd say if you gave me ten of these to do mentally I'd get maybe 2 correct.

For me:

3974 * 438 -> 4000 * 438 - 26*438 ->
4000 * 438 - 26*440 - 26*2 -> 4000 * 438 - 20*440 - 6*440 - 26*2

And so on, and I'd do some of the intermediate calculations as I go (e.g. 20*440 and 6*440).

But that's only really needed if I need a precise answer. If I can get away with an estimate, I'll simplify it even more:

4000 * 430 ~= 43 * 4 * 10000 = 86 * 2 + 10000 = 1,720,000

Actual answer: 1,740,612. 4000 * 440 would be easier (I like multiplying 4s), but I know it would overshoot, so I round one up and the other down. Close enough for something like estimating how much a large quantity of something kind of expensive would cost (i.e. if my company gave everyone a hot tub or something).

Depends how much neuron density you have in the part of the brain that handles this. It's mostly about memory, being able to accurately and quickly remember all the little steps you have already done and what the results of those steps were. Then just keep going one digit pair at a time keeping in mind all the results so you can deal with the carry overs.

But the whole reason we can focus on teaching everyone shortcuts for smaller math now is because we do literally always have a calculator on us now. So while it's still good to know how to do bigger math more efficiently, you'll never catch up to a calculator anymore. It's more important that they know the foundational concept well enough to move on to the next step now rather than practicing doing big math faster and faster. Can leave that to the individuals with talent in the area.