SOLUTION: Simplify: ((x-2)/(x+5)) - ((x^2-2x-3)/(x^2-x-6)) x ((x^2+2x)/(x^2-4x))

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Question 1189207: Simplify:
((x-2)/(x+5)) - ((x^2-2x-3)/(x^2-x-6)) x ((x^2+2x)/(x^2-4x))
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!

%28%28x%5E%22%22+-+2%29%2F%28x%5E%22%22+%2B+5%29%29%22%22-%22%22

Factor all the fractions on the right:

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22

Cancel the (x-3)'s

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22%28%28x%2B1%29%2F%28x%2B2%29%29%2A%28%28x%28x%2B2%29%29%2F%28x%28x-4%29%29%29

Cancel the x's

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22%28%28x%2B1%29%2F%28x%2B2%29%29%2A%28%28x%2B2%29%2F%28x-4%29%29

Cancel the (x+2)'s

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22%28%28x%2B1%29%2F%28cross%28x%2B2%29%29%29%2A%28%28cross%28x%2B2%29%29%2F%28x-4%29%29

%28%28x+-+2%29%2F%28x+%2B+5%29%29%22%22-%22%22%28%28x%2B1%29%2F%28x-4%29%29

The LCD is (x+5)(x-4), so multiply the fraction on the left 
by %28%28x-4%29%2F%28x-4%29%29, which just equals 1.  Also, multiply 
the fraction on the right by %28%28x-5%29%2F%28x-5%29%29, which also 
just equals 1: 

%28%28x+-+2%29%2F%28x+%2B+5%29%29%28%28x-4%29%2F%28x-4%29%29%22%22-%22%22%28%28x%2B1%29%2F%28x-4%29%29%28%28x%2B5%29%2F%28x%2B5%29%29

Indicate the multiplication of numerators and of denominators 
in both terms:

%28%28x+-+2%29%28x+-+4%29%29%2F%28%28x%2B5%29%28x-4%29%29%22%22-%22%22%28%28x%2B1%29%28x%2B5%29%29%2F%28%28x-4%29%28x%2B5%29%29

Use "FOIL" to multiply the numerators ONLY, but leave the 
denominators factored:

%28%28x%5E2+-+6x+%2B8%29%29%2F%28%28x%5E%22%22%2B5%29%28x%5E%22%22-4%29%29%22%22-%22%22%28%28x%5E2%2B6x%2B5%29%29%2F%28%28x%5E%22%22-4%29%28x%5E%22%22%2B5%29%29

The denominators are the same, so we subtract the numerators 
and place them over the common denominator:



Remove the parentheses, being careful to change the
signs in the second parenthetical expression since 
it is preceded by a - sign:

%28x%5E2+-+6x+%2B8-x%5E2-6x-5%29%2F%28%28x%5E%22%22%2B5%29%28x%5E%22%22-4%29%29

Combine like terms:

%28-12x+%2B3%29%2F%28%28x%2B5%29%28x-4%29%29

We take common factor -3 out of the numerator:

%28-3%284x-1%29%29%2F%28%28x%2B5%29%28x-4%29%29

Edwin