For most practical think on your feet applications, multiplying numbers should be done by rounding them to close approximations. So you have 357 and 289. 357 is close to 350. 289 is close to 300, the difference of rounding is good because you're going up in one and down in another. So now you're looking at multiplying 300x350. 100x100 is 10,000, so 3*3 =9, so 300x300 is 90,000. now you just have the trivial matter of 300x50. 3/2=1.5, so you're looking at 10.5 * 10,000. So you have the answer is 10500. With the aid of a calculator, I got 103,173 so it isn't far off. It is real easy to think about this way.
If you think on your feet and you try to do stuff like,"Carry the 1, remember a number in a short term memory cache, and then do another multiplication and try and remember again, by the time you do your new calculation, you probably forgot your short term." I'd hazard a guess(this is all conjecture from now on) that brains like to use the same variable over and over again to store data, and when you start doing a complex calculation it could rewrite the same memory location. I don't know about your brain, but I budget between processing and memory. If I'm constantly thinking about stuff, my memory goes and I need an external aid. If I'm just memorizing stuff, my processing goes. Try doing 357x289 and use a notepad to record your states and it is easy(memorization). Alternatively you could use a calculator and do 357x9 then memorize the result, followed by 357x80 then memorize result followed by 357x300 and memorize the result. Then you use your memory to add the three numbers in your head. If you can be pure memory or processing, the task doesn't get difficult.