What is the product? x^2-16/2x+8 times x^3-2x^2+x/x^2+3x-4

Step-by-step explanation:

Rewrite x^2-16/2x+8 as

x^2 - 16

2(x + 8)

and rewrite x^3-2x^2+x/x^2+3x-4 as

x^3-2x^2+x

  x^2+3x-4

Next, write "x^2-16/2x+8 times x^3-2x^2+x/x^2+3x-4" as

x^2 - 16              x^3-2x^2+x

     *

 2(x+8)                  x^2+3x-4

Now factor both numerators and both denominators.  We get:

(x - 4)(x + 4)(x)(x^2 - 2x + 1)

   2(x + 8)(x + 4)(x - 1)

Now begin cancelling wherever possible.  The x + 4 factors cancel, leaving us with:

(x - 4)(x)(x - 1)(x - 1)

   2(x + 8)(x - 1)

Cancelling the x - 1 factors yields:

(x - 4)(x)(x - 1)           x(x - 4)

=

  2(x + 8)                 2(x + 8)

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