Please help!!
Simplify the following rational expression. List all of the excluded values, and classify each as a removable discontinuity, or a nonremovable discontinuity.
-2/x^2-4 + x-1/x^2-2x

Step-by-step explanation:

Identify: the problem is asking you to do three things:

Simplify the expression

Find the excluded values (points of discontinuity)

Classify each excluded value as a removable discontinuity or nonremovable discontinuity.

Think: Removable discontinuities are roots of factors that appear in both the numerator and denominator.  Nonremovable discontinuities are roots of factors that only appear in the denominator.  So to find the excluded values, we must first combine these two fractions into one by finding the common denominator.

Do:

-2 / (x² − 4) + (x − 1) / (x² − 2x)

First, factor the denominators.

-2 / ((x − 2) (x + 2)) + (x − 1) / (x (x − 2))

The common denominator is x (x − 2) (x + 2).

-2x / (x (x − 2) (x + 2)) + (x + 2) (x − 1) / (x (x − 2) (x + 2))

Add the fractions.

(-2x + (x + 2) (x − 1)) / (x (x − 2) (x + 2))

Simplify the numerator.

(-2x + x² + x − 2) / (x (x − 2) (x + 2))

(x² − x − 2) / (x (x − 2) (x + 2))

Factor the numerator.

(x − 2)(x + 1) / (x (x − 2) (x + 2))

There are three factors in the denominator, so there are three excluded factors.  x = 0, x = 2, and x = -2.

Since x − 2 appears in the numerator, x = 2 is a removable discontinuity.

x = 0 and x = -2 are nonremovable discontinuities.

(x − 2)(x + 1) / (x (x − 2) (x + 2))

x = -2 is a nonremovable discontinuity.

x = 0 is a nonremovable discontinuity.

x = 2 is a removable discontinuity.