Question 935835

This question is from my ACT Math Study book. I'm having trouble understanding the steps to find the answer. 

The Question says:
Simplify: ((x^2-9)/(x+1)) / ((x+3)/(x^2-1))
(its written as a fraction divided by the second as a fraction)

I know the first thing to do is flip the second fraction and then multiply, so I did this and got: (x^2-9)(x^2-1) over (x+1)(x+3). I then FOILed the top and bottom to get: (x^4-10x^2+9) over (x^2 +4x +3). Then I thought I could reduce to: x^4-9x^2+6 over 4x. This answer is wrong. 

The solution in the book is exactly this:
((x^2-9)/(x+1)) / ((x+3)/(x^2-1)) = ((x+3)(x-3)/(x+1))*((x+1)(x-1)/(x+3))=(x-3)(x-1)=x^2-4x+3. 

I'm not following the steps that the book used to come up with this answer, and am wondering why my approach is incorrect. Thank You for your help!
<pre><font face = "Tohoma" size = 4 color = "indigo"><b>{{{((x^2 - 9)/(x + 1))/((x+3)/(x^2-1))}}}
The easiest and LEAST confusing way to continue with this is to separate the 2 fractions as follows:
{{{(x^2 - 9)/(x + 1)}}} ÷ {{{(x + 3)/(x^2 - 1)}}} ------ I'm sure this looks a lot less confusing now!!
{{{((x - 3)(x + 3))/(x + 1)}}} ÷ {{{(x + 3)/((x - 1)(x + 1))}}} ----- Factoring {{{x^2 - 9}}} and {{{x^2 - 1}}}
{{{((x - 3)(x + 3))/(x + 1)}}} * {{{((x - 1)(x + 1))/(x + 3)}}} ----- Changing DIVISION SYMBOL to MULTIPLICATION, and then INVERTING DIVISOR
{{{((x - 3)cross((x + 3)))/cross((x + 1))}}} * {{{((x - 1)cross((x + 1)))/cross((x + 3))}}} --- Cancelling numerators and denominators
{{{(x - 3)(x - 1)}}}, or {{{highlight_green(x^2 - 4x + 3)}}}
You can do the check!! 
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