Simplify ( 2x^2 - 8 )/( x - 2 ) + 13/x( x -2 ) - ( 4x^2 - 16 )/( x + 2 )

In order to calculate the sum or difference of 3 ratios,
we have to verify if they have a common denominator.

But,
before verifying if they have a common denominator, we'll solve the difference of
squares from the brackets.

We'll factorize by 2 the first
ratio:

( 2x^2 - 8 )/( x - 2 ) = 2(x^2 - 4)/( x - 2
)

We'll write the difference of squares (x^2 - 4) as a
product:

(x^2 - 4) =
(x-2)(x+2)

We'll re-write the
ratio;

( 2x^2 - 8 )/( x - 2 ) = 2(x-2)(x+2)/(x
- 2)

We'll reduce like
terms:

 ( 2x^2 - 8 )/( x - 2 ) =
2(x+2)

We'll factorize by 4 the third
ratio:

( 4x^2 - 16 )/( x + 2 ) = 4(x^2 - 4)/( x + 2
)

4(x^2 - 4)/( x + 2 )
=  4(x-2)(x+2)/(x + 2)

We'll reduce like
terms:

4(x^2 - 4)/( x + 2 ) =
4(x-2)

We'll re-write now the given
expression:

2(x+2) + 13/x( x -2 ) -
4(x-2)

It's obvious that the least common denominator (LCD)
is the denominator of the second ratio:

To calculate the
expression we'll do the steps:

- we'll multiply 2(x+2) by
x*(x-2)

- we'll multiply 4(x-2) by
x*(x-2)

We'll
get:

2(x+2)*x*(x-2) + 13
-  4(x-2)*x*(x-2)

We'll open the
brackets:

2x^3 - 8x + 13 - 4x^3 + 16x^2 -
16x

We'll group like terms and we'll
get:

-2x^3 + 16x^2 - 24x +
13