Check the FOIL examples then.

Carrying is due to the number base you’re working in… in this case you’re using base 10.

Remember: at the end of the day you’re saying you’ve got 51 groups each with 52 elements and you want to add them all up. The rest is just different ways of arranging those elements into groups that are easier to work with.

(This is MUCH harder to explain over text...way easier to show you...I’ll do my best)

This is a way to break down large number multiplication into single digit multiplication.

The first term we have is 51. So I decided to put that on top.
The second term we have is 52. So I put that under it.
That looks like this:
51
52
—

Now I can multiply the individual numbers together, but I have to make sure I write them down in the right order.

So, first we multiply the top term by the first digit of the bottom term (a 5)^((from 52 into both the 5 and 1 from 51)).

Ok, what’s 5x5?
25

What’s 5x1?
5

So the first two get written on the same line right?
That gives us 25 in the 10s spot and 5 in the 1s spot. (The 2 carrying over to the 100s spot automatically). (So we have 250 + 5 to break it ALL the way down) or more simply: 255.

Now what’s 2x5?
10

And 2x1?
2

When we move to the next digit in the second term we also need to move to a new line. But remember that the next digit (the 2) is one place to the right of the previous digit. So we need to move our answers one digit to the right as well.
Now we take the 10 in the 10s place (letting it’s 1 fall in the hundreds place.) and we set the 2 in the 1s place. (To break it down again we get 100 + 2) or more simply 102.

Now we have 255 and 102 right? And 255 goes down on the first line, with 102 on the second, but one place to the right.

That looks like this:
255
_ 102
Right?

So the 2 drops down and adds with the nothing for a 2.
2- - -
Then the 5 drops down and adds with the 1 for 6.
2 6 - -
Then the next 5 drops and adds with the 0 for 5.
2 6 5 -
We’ve run out of numbers on the first line, but have one more on the next. We’ll drop a 0 (to fill that place) and add it to the 2 we have left for 2.
2 6 5 2
And that’s our answer.

So, to write it out again:
51
x52
——
2 5 5
+ 1 0 2
————
2 6 5 2

This works for any size number multiplied with any other number of any size.