Question

Subtract the rational expressions and simplify: (1)/(x²+6x+8) − (1)/(x²-2x−8).
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Answer 1

To subtract the rational expressions and simplify, we need to find a common denominator and then combine the numerators.

Step-by-step explanation:

To subtract the rational expressions and simplify, we need to find a common denominator. The denominators are both quadratic expressions, so we need to factor them.

The first denominator, x²+6x+8, factors as (x+4)(x+2).

The second denominator, x²-2x-8, factors as (x-4)(x+2).

Now we can rewrite the expression with the common denominator:

(1)/(x+4)(x+2) - (1)/(x-4)(x+2)

To subtract the fractions, we need to multiply the first fraction by (x-4)/(x-4) and the second fraction by (x+4)/(x+4).

Now we can combine the numerators:

(x-4) - (x+4)/(x+4)(x-4)

Simplifying the numerator, we get -8/(x+4)(x-4), which is the final simplified expression.