Question 198535
<pre><font size = 4 color = "indigo"><b>
{{{((a-2)/(a+2) + (a-2)) / (a - (3a+12)/(a+2))}}}

Put a {{{1}}} under the {{{(a-2)}}} in the top and
under the {{{a}}} on the bottom so everything
will be a fraction:

{{{((a-2)/(a+2) + (a-2)/1) / (a/1 - (3a+12)/(a+2))}}}

Enclose every fraction within parentheses:

{{{(((a-2)/(a+2))+((a-2)/1))/((a/1) - ((3a+12)/(a+2)))}}}

Since the LCD for all four fractions top and bottom
is {{{(a+2)}}}, multiply top and bottom by {{{(a+2)}}}
written as {{{((a+2)/1)}}}

{{{(   

((a+2)/1) ((a-2)/(a+2)) 

+ 

((a+2)/1)((a-2)/1)

) 

/ 

(   

((a+2)/1)(a/1) - ((a+2)/1)((3a+12)/(a+2))

)

}}}

Cancel the {{{(a+2)}}}'s in two places:

{{{(   

((cross(a+2))/1) ((a-2)/(cross(a+2))) 

+ 

((a+2)/1)((a-2)/1)

) 

/ 

(   

((a+2)/1)(a/1) - ((cross(a+2))/1)((3a+12)/(cross(a+2)))

)

}}}

Ignoring all the {{{1}}} denominators, all we have
left is

{{{
(
(a-2)+(a+2)(a-2)
)
/
(
(a+2)a-(3a+12)
)
}}}

Use FOIL to change {{{(a+2)(a-2)}}} to {{{a^2-2a+2a-4}}} 
to {{{a^2-cross(2a)+cross(2a)-4}}} to {{{a^2-4}}}
and multiply out the bottom:


{{{
(
(a-2)+(a^2-4)
)
/
(
a^2+2a-3a-12
)
}}}

{{{
(
a-2+a^2-4
)
/
(
a^2-a-12
)
}}}

{{{
(
a^2+a-6
)
/
(
a^2-a-12
)
}}}

Factor the numerator and denominator:

{{{(
(a+3)(a-2)
)/
(
(a-4)(a+3))
}}}

Cancel the {{{(a+3)}}}'s

{{{(
(cross(a+3))(a-2)
)/
(
(a-4)(cross(a+3)))
}}}

All that's left is

{{{(a-2)/(a-4)}}}

Edwin</pre>