Question 734748
<pre>
       {{{((2x+12)/(3x-15))/((2x^2-6x-20)/(x^2-10x+25))}}}

Write as a division of fractions:

       {{{(2x+12)/(3x-15)}}}{{{"÷"}}}{{{(2x^2-6x-20)/(x^2-10x+25)}}}

2x+12 factors as 2(x+6)
3x-15 factors as 3(x-5)
2x²-6x-20 factors as 2(x²-3x-10) or as 2(x-5)(x+2)
x²-10x+25 factors as (x-5)(x-5)

       {{{(2(x+6))/(3(x-5))}}}{{{"÷"}}}{{{(2(x-5)(x+2))/((x-5)(x-5))}}}
 
Invert the second franction and change division to multiplication

       {{{(2(x+6))/(3(x-5))}}}{{{""*""}}}{{{((x-5)(x-5))/(2(x-5)(x+2))}}}

Cancel the 2's

       {{{(cross(2)(x+6))/(3(x-5))}}}{{{""*""}}}{{{((x-5)(x-5))/(cross(2)(x-5)(x+2))}}}

       {{{((x+6))/(3(x-5))}}}{{{""*""}}}{{{((x-5)(x-5))/((x-5)(x+2))}}}

Cancel one pair of (x-5)'s

       {{{((x+6))/(3(cross(x-5)))}}}{{{""*""}}}{{{((cross(x-5))(x-5))/((x-5)(x+2))}}}

       {{{(x+6)/3}}}{{{""*""}}}{{{(x-5)/((x-5)(x+2))}}}

Cancel the other pair of (x-5)'s

       {{{(x+6)/3}}}{{{""*""}}}{{{(cross(x-5))/((cross(x-5))(x+2))}}}

       {{{(x+6)/(3(x+2))}}}

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

         {{{(3x)/(x^2+3x-10)}}}{{{""-""}}}{{{(2x)/(x^2+x-6)}}}
          
         {{{(3x)/(x+5)(x-2))}}}{{{""-""}}}{{{(2x)/((x+3)(x-2))}}} 

Get LCD of (x+5)(x-2)(x+3)

         {{{(3x(x+3))/(x+5)(x-2)(x+3))}}}{{{""-""}}}{{{(2x(x+5))/((x+3)(x-2)(x+5))}}}

         {{{(3x^2+9x)/(x+5)(x-2)(x+3))}}}{{{""-""}}}{{{(2x^2+10x)/((x+3)(x-2)(x+5))}}}

The denominators are equal, so we combine the numerators over the LCD

         {{{((3x^2+9x)-(2x^2+10x))/((x+5)(x-2)(x+3))}}}

         {{{(3x^2+9x-2x^2-10x)/((x+5)(x-2)(x+3))}}}

         {{{(x^2-x)/((x+5)(x-2)(x+3))}}}

         {{{(x(x-1))/((x+5)(x-2)(x+3))}}}

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

          {{{(2x-10)/(2x^2-5x-3)}}}{{{""+""}}}{{{1/(x-3)}}}{{{""=""}}}{{{3/(x-3)}}}

          {{{(2x-10)/((x-3)(2x+1))}}}{{{""+""}}}{{{1/(x-3)}}}{{{""=""}}}{{{3/(x-3)}}}

Multiply through by the LCD of (x-3)(2x+1)  

          2x-10 + 1(2x+1) = 3(2x+1)
         2x - 10 + 2x + 1 = 6x + 3
                   4x - 9 = 6x + 3
                      -2x = 12
                        x = -6                        


Edwin</pre>