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
