[multiplying rational expressions][section 53]

(1.)  3x - 48y2       3x + 6y
     ___________ * ____________           here is the problem
      2x2 - 8y2       3x + 12y

=    3(x - 16y2)             3(x + 2y)        
    _________________  *   _______________    factor
     2(x2 - 4y2)              3(x + 4y)

=     3(x - 16y2)                3(x + 2y)
    ______________________  *  _______________
        2(x - 2y)(x + 2y)        3(x + 4y)      factor

=        3(x - 16y2)
    __________________________   cancel and multiply
       2(x - 2y)(x + 4y)


(2.)  x2 + 8x + 16     x - 3
      _____________ * ________          here is the problem
         x2 - 9        x + 4

=   (x + 4)2                (x - 3)        factor
    ________________  *   ____________
    (x - 3)(x + 3)         (x + 4)

=      (x + 4)
    ___________________          cancel and multiply
       (x + 3)

(3.)    9 - x2           x
        ______  *   _________            here is the problem
        x + 3         3 - x


=    (3 - x)(3 + x)             x
   ___________________  * _______________
          (x + 3)           -(x - 3)        factor

=    -x                 cancel and multiply

(4.)     24x2                3x - 6
      ________________  *  _________         here is the problem
      3(x2 - 4x + 4)          2x

=      24x2             3(x - 2)
    _______________ *  ____________
      3(x - 2)2            2x           factor

=    12x/(x - 2)         cancel and multiply

(5.)   5m + 5n        m2 - mn
       _________  *  ____________      here is the problem
       m2 - n2         (m + n)2   

=     5(m + n)               m(m - n)
    ____________________  *  _______________
    (m + n)(m - n)           (m + n)2          factor

=      5m/(m + n)2            cancel and multiply


(6.)    a2 + 6a + 5         5a - 15
       _____________  * _______________   here is the problem
           a - 3         a2 + 4a - 5


=    (a + 5)(a + 1)          5(a - 3)
    __________________ *  ________________
          (a - 3)          (a + 5)(a - 1)   factor

=      5(a + 1)/(a - 1)    cancel and multiply

(7.)   x2 - 3x - 18          3x + 3
      ________________ *  ______________   here is the problem
        x2 - x - 2          x2 - 2x - 15

=    (x - 6)(x + 3)             3(x + 1)
   ___________________   *  ___________________
    (x + 1)(x - 2)          (x - 5)(x + 3)      factor

=    3(x - 6)
     _______________
     (x - 2)(x - 5)     cancel and multiply

(8.)     4x2 - 9y2                    6xy     here is the problem
      ____________________  *  ________________
          6x2 - 9xy                4xy + 6y2  

 =     (2x - 3y)(2x + 3y)               6xy
   __________________________  *  __________________
       3x(2x - 3y)                   2y(2x + 3y)  factor

=      1               cancel

(9.)   a2 + a - 2             a2 - 2a - 8     here is the problem
      ________________  *   ________________
        a2 - 3a - 10          a2 - 7a + 12

=     (a + 2)(a - 1)           (a - 4)(a + 2)
    ______________________ *  _________________
      (a - 5)(a + 2)          (a - 3)(a - 4)    factor

=    (a - 1)(a + 2)
     _________________
     (a - 5)(a - 3)         cancel and multiply



(10.)   c2 - 6c               c + 3
        ____________   *   ____________     here is the problem
          c - 6                  c

=   c(c - 6)        (c + 3)
   ____________ *  ____________      factor
    (c - 6)            c

=     c + 3               cancel and multiply