Final answer:
The expression x+4/x^2−9x+20−x/x^2−2x−15 is equivalent to (8x^2 - 71x + 80)/((x^2 - 9x + 20)*(x^2 - 2x - 15)).
Step-by-step explanation:
The given expression is:
x + 4 / (x^2 - 9x + 20) - x / (x^2 - 2x - 15)
To simplify the expression, we can find a common denominator for both fractions. The common denominator for the two fractions is (x^2 - 9x + 20) * (x^2 - 2x - 15).
After finding the common denominator, we can combine the fractions by multiplying the numerator of the first fraction by (x^2 - 2x - 15) and the numerator of the second fraction by (x^2 - 9x + 20). This gives us:
(x * (x^2 - 2x - 15) + 4 * (x^2 - 9x + 20) - x * (x^2 - 9x + 20)) / ((x^2 - 9x + 20) * (x^2 - 2x - 15))
Expanding and simplifying the expression, we get:
(x^3 - 2x^2 - 15x + 4x^2 - 36x + 80 - x^3 + 9x^2 - 20x) / ((x^2 - 9x + 20) * (x^2 - 2x - 15))
Combining like terms, we have:
(8x^2 - 71x + 80) / ((x^2 - 9x + 20) * (x^2 - 2x - 15))
Therefore, the expression is equivalent to (8x^2 - 71x + 80) / ((x^2 - 9x + 20) * (x^2 - 2x - 15)).