4r²-20ry+25y²


Multiply the 4 by the 25 ignoring signs.  Get 100

Write down all the ways to have two positive integers
which have product 100, starting with 100*1

100*1
 50*2
 25*4
 20*5
 10*10
 

Since the last sign in 4r²-20ry+25y² is +, ADD them,
and place the SUM out beside that:



100*1    100+1=101
 50*2     50+2=52
 25*4     25+4=29
 20*5     20+5=25
 10*10   10+10=20


Now, again ignoring signs, we find in that list of
sums the coefficient of the middle term in 4r²-20ry+25y²

So we replace the number 20 by 10+10

4r²-20ry+25y²
4r²-(10+10)ry+25y²

Then we distribute to remove the parentheses:

4r²-10ry-10ry+25y²

Factor the first two terms 4r²-10ry by taking out the
greatest common factor, getting 2r(2r-5y)

Factor the last two terms -10ry+25y² by taking out the
greatest common factor, -5y, getting -5y(2r-5y)

So we have

2r(2r-5y)-5y(2r-5y)
Notice that there is a common factor, (2r-5y)

2r(2r-5y)-5y(2r-5y)

which we can factor out leaving the 2r and the -5y to put 
in parentheses:

(2r-5y)(2r-5y)

But since the two factors are the same, we can write it
simply as

(2r-5y)²

Edwin