Arianna needed 140x²+60x square inches of paper to make a book 
jacket 10x inches tall. In figuring the area she needed, she 
allowed for a front and back flap. If the spine of the book 
jacket is 2x inches, and the front and back of the book jacket 
are 6x inches, how wide are the front and back flaps? A quotient 
of polynomials can be used to identify the answer.

(Area) = (length)(width)

(Area) = (140x²+60x)

(length) = the horizontal (left-to-right) dimension 
         = (f+6x+2x+6x+f) =
         = (combine like terms) = (2f+14x)

width = the vertical (top to bottom) dimension = (10x)

Substitute in 

         Area = (length)(width)
  (140x²+60x) = (2f+14x)(10x)

Reverse the factors on the right to make it easier to 
distribute:

  (140x²+60x) = 10x(2f+14x)
    140x²+60x = 20xf+140x²

Subtract 140x² from both sides

          60x = 20xf

Divide both sides by 20x

         

divide by canceling:

         

         

The front and back flaps must be 3 inches wide. 

[Not much division of polynomials is required. Just division
by canceling terms in numerator and denominator.]

Edwin