SOLUTION: <pre>Please show me how to do long division this way! x² - 49y² ——————————— ÷ (x² - 7xy) 6x² + 42y Thanks!</pre>

Please show me how to do long division this way! 

  x² - 49y²
——————————— ÷ (x² - 7xy)
 6x² + 42y

This isn't long division.  It's simplifying by 
"factoring and cancelling":

  x² - 49y²
——————————— ÷ (x² - 7xy)
 6x² + 42y

Write the second fraction over 1

  x² - 49y²    x² - 7xy 
——————————— ÷ ——————————
 6x² + 42y        1

Invert the second fraction and change to multiplication:

  x² - 49y²       1
——————————— × ——————————
 6x² + 42y     x² - 7xy   


Factor the upper left numerator: x² - 49y² = (x - 7y)(x + 7y)
Factor the lower left denominator: 6x² + 42y = 6(x² + 7y)
Factor the lower right denominator: x² - 7xy = x(x - 7y)


 (x - 7y)(x + 7y)         1
—————————————————— × ——————————
    6(x² + 7y)        x(x - 7y)

Indicate the multiplication of the numerators and 
the denominators and write as a single fraction:

   (x - 7y)(x + 7y)     
——————————————————————
  6(x² + 7y)x(x - 7y)

Write the x next to the 6 as it is customary to write
shorter factors first:


   (x - 7y)(x + 7y)     
——————————————————————
  6x(x² + 7y)(x - 7y)

Cancel the (x - 7y)'s

      1
   (x - 7y)(x + 7y)     
——————————————————————
  6x(x² + 7y)(x - 7y)
                1

   (x + 7y)     
——————————————
  6x(x² + 7y)

Caution: Do not try to cancel the remaining parenthetical
factors, because they are not exactly alike, since the x is
squared in the bottom but not in the top and you cannot cancel
terms between numerator and denominator.  So the only thing
you can do is erase the parentheses in the top:

    x + 7y     
——————————————
  6x(x² + 7y)


Edwin
AnlytcPhil@aol.com