Question 1186675
<pre>
He's right about the parentheses when typing numerators and denominators all
on one line. Otherwise, you can't tell where a numerator or denominator starts
and ends.  Here is how you should have typed your problem:

((9x^2 - 64)/(x^2+7x+12)) ÷ ((3x^2+17x+24)/(x^2+6x+9))

Correct punctuation is even more important in math than in English. Anyways,
I'll assume what I think you meant to be a numerator and a denominator, and
where I think you meant for them to be.

----------------------------------------------------------

Even after we simplify the expression, we must make restrictions to make sure
that there are no numbers we could substitute for x that would cause the
original expression to be undefined by dividing by 0. 

Therefore, 
EACH TIME WE CANCEL ANY FACTOR, WE MUST MAKE A RESTRICTION THAT PREVENTS THE
CANCELED EXPRESSION FROM EVER BEING EQUAL TO ZERO.  ALSO IN THE FINAL
SIMPLIFICATION, WE MUST MAKE A RESTRICTION THAT PREVENTS ITS DENOMINATOR
FROM BEING EQUAL TO ZERO.

{{{(9x^2 - 64)/(x^2+7x+12)}}} ÷ {{{(3x^2+17x+24)/(x^2+6x+9)}}} 

Invert and multiply:

{{{((9 x^2 - 64)/(x^2 + 7 x + 12))}}}{{{""*""}}}{{{((x^2 + 6 x + 9)/(3 x^2 + 17 x + 24))}}}

Factor:
{{{((3x-8)(3x+8)/(x+4)(x+3))}}}{{{""*""}}}{{{((x+3)(x+3)/(x+3)(3x+8))}}}

We cancel the (3x+8)'s
{{{3x+8<>0}}
{{{3x<>-8}}}
{{{x<>-8/3}}}

We must indicate the restriction that x cannot equal -8/3.

{{{((3x-8)(cross(3x+8))/(x+4)(x+3))}}}{{{""*""}}}{{{((x+3)(x+3)/(x+3)(cross(3x+8)))}}}

We cancel one of the (x+3)'s and indicate the restriction
{{{x+3<>0}}}
{{{x<>-3}}}

{{{((3x-8)(cross(3x+8))/(x+4)(cross(x+3)))}}}{{{""*""}}}{{{((cross(x+3))(x+3)/(x+3)(cross(3x+8)))}}}

We cancel the other pair of (x+3)'s.  [We've already indicated this restriction,
so we don't need to indicate it again.]

{{{((3x-8)(cross(3x+8))/(x+4)(cross(x+3)))}}}{{{""*""}}}{{{((cross(x+3))(cross(x+3))/(cross(x+3))(cross(3x+8)))}}}

The final simplification is

{{{(3x-8)/(x+4)}}}

We must also indicate that the denominator of the final simplification to
prevent it from ever being 0.

{{{x+4<>0}}}
{{{x<>-4}}}

So putting in all the restrictions:

{{{matrix(1,7,
(3x-8)/(x+4),",",
x<>-8/3,",", x<>-3,",", x<>-4 )}}}

Edwin</pre>