Calculate (x-2)/(x+1)+(x+3)/(x+2)-(5-x^2)/(x^2+3x+2).

To perform additions and subtractions of ratios, they must
have the same denominator.

We'll have to calculate the
least common denominator of the given ratios. For this reason, we'll use factorization
for the denominator of the 3rd ratio.

We'll determine the
roots of the expression x^2+3x+2:

x^2+3x+2 =
0

We'll apply the quadratic
formula:

x1 = [-3 + sqrt(9 -
8)]/2

x1 = (-3+1)/2

x1 =
-1

x2 = (-3-1)/2

x2 =
-2

We'll re-write the expression x^2+3x+2 as a product of
linear factors:

x^2+3x+2 =
(x-x1)(x-x2)

x^2+3x+2 =
(x+1)(x+2)

We'll re-write the given
expression:

E(x) =
(x-2)/(x+1)+(x+3)/(x+2)-(5-x^2)/(x+1)(x+2)

We notice that
the least common denominator of the 3 ratios is the denominator of the 3rd
ratio:

LCD = (x+1)(x+2)

E(x) =
[(x-2)(x+2)+(x+3)(x+1)-(5-x^2)]/(x+1)(x+2)

We'll remove the
brackets. We'll recognize the difference of
squares:

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

E(x) =
[(x-2)(x+2)+(x+3)(x+1)-(5-x^2)]/(x+1)(x+2)

E(x) = (x^2 - 4
+ x^2 + 4x + 3 - 5 + x^2)/(x+1)(x+2)

We'll combine like
terms:

E(x) = (3x^2 + 4x -
6)/(x+1)(x+2)