SOLUTION: Can someone factor this please? 36x2 − 97x + 36

36x² - 97x + 36

Multiply the 36 by 36, disregarding signs, getting 1296.

Write down a list of all the ways to factor 1296 using two 
positive integers:

1296*1
 648*2
 432*3
 324*4
 216*6
 162*8
 144*9
 108*12
  81*16
  72*18
  54*24
  48*27
  36*36

Now look at the sign of the last term to see whether we add or
subtract those pairs of numbers.  The last term is +36, which
has a + sign, so we add those same pairs of numbers that
we multiplied above to get 1296. 
(when the last term has a - sign, we subtract, but since it has a +, we add):


1296+1  = 1297
 648+2  =  650
 432+3  =  435 
 324+4  =  328
 216+6  =  222
 162+8  =  170
 144+9  =  243
 108+12 =  120
  81+16 =   97
  72+18 =   90
  54+24 =   78
  48+27 =   75 
  36+36 =   72

Next we look down that list to find the coefficient of
the middle term of 36x² - 97x + 36, in absolute value,
which is 97 and we find it as

  81+16 =   97

So we substitute (81+16) in place of 97

  36x² - 97x + 36

36x² - (81+16)x + 36

and remove the parentheses:

36x² - 81x - 16x + 36

Next we factor only the FIRST two terms by
factoring out 9x:

9x(4x - 9) - 16x + 36

Next we factor only the LAST two terms by
factoring out -4:

9x(4x - 9) - 4(4x - 9)

Notice that the two parentheses are the same.
I'll color them red:

9x(4x - 9) - 4(4x - 9)

So factor out the red factor, leaving the black factors as terms in 
parentheses:

(4x - 9)(9x - 4)

Edwin