Answer: To find the difference between the expressions 4/x^2 + 5 and 1/x^2 - 25, we need to subtract the second expression from the first.
Given:
Expression 1: 4/x^2 + 5
Expression 2: 1/x^2 - 25
To subtract these expressions, we need a common denominator. The common denominator in this case is x^2(x^2 - 25), which is the least common multiple of the denominators.
Now, let's perform the subtraction:
(4/x^2 + 5) - (1/x^2 - 25)
To subtract the fractions, we need to have the same denominator for both terms:
[(4(x^2 - 25))/(x^2(x^2 - 25))] + [(5x^2)/(x^2(x^2 - 25))] - [(1(x^2))/(x^2(x^2 - 25))] + [(25(x^2))/(x^2(x^2 - 25))]
Combining the terms over the common denominator:
[(4x^2 - 100 + 5x^2 - x^2 + 25x^2)] / (x^2(x^2 - 25))
Simplifying the numerator:
(4x^2 + 5x^2 - x^2 + 25x^2 - 100) / (x^2(x^2 - 25))
(34x^2 - 100) / (x^2(x^2 - 25))
Therefore, the difference between the expressions 4/x^2 + 5 and 1/x^2 - 25 is (34x^2 - 100) / (x^2(x^2 - 25)).