Question 464931
{{{ (x^2 - 16) / (x^2 + 6x + 8) }}}
Anything that looks like this, usually you
must factor the top, bottom, or both.
Here you must factor both top and bottom
--------------------------------
The top is the difference of 2 squares, and the factors
always look like {{{ (x + 4)*(x -4) }}} because the 
middle term gets subtracted out like this:
{{{ (x + 4)*(x - 4) = x^2 + 4x - 4x - 16 }}}
---------------------------------
A good way to factor the bottom is to " complete
the square"
First set the expression equal to zero,
Then subtract {{{8}}} from both sides
Take 1/2 the coefficient of {{{x}}}, square it, and
add it to both sides
{{{ x^2 + 6x  + (6/2)^2 = -8 + (6/2)^2 }}}
{{{ x^2 + 6x + 9 = -8 + 9 }}}
{{{ (x + 3)^2 = 1 }}}
Take the square root of both sides
{{{ x + 3 = 1 }}}
{{{ x + 2 = 0 }}}
and
{{{ x + 3 = -1 }}}
{{{ x + 4 = 0 }}}
The factors are {{{ x + 2 }}} and {{{ x + 4 }}}
Now the original fraction looks like:
{{{ ((x + 4)*(x - 4)) / ((x + 2)*(x + 4)) }}}
This is the same as:
{{{ ((x + 4)/(x + 4))* ((x-4)/(x+2)) = (x-4)/(x+2) }}}
-------------------------------------
The 2nd problem has the difference of 2 squares on the bottom
You can factor the top just as in the 1st problem.
The bottom is: {{{ x^2 - 25 = (x+5)*(x-5) }}}
When you complete the squares, the top is:
{{{  x^2 - 3x - 10 = (x - 5)*(x + 2) }}}
Now you have:
{{{ ((x - 5)/(x - 5))*((x + 2)/(x + 5)) = (x + 2)/(x + 5) }}}