Answer:
x^3 + 5
Explanation:
To simplify the expression x+1/x + (x^2+5x)/(x^2+4x-5), we can follow these steps:
Step 1: Simplify the first term, x + 1/x. To do this, we need to find a common denominator for the terms.
The common denominator is x, so we can rewrite the expression as (x^2/x) + (1/x).
Combining the terms, we get (x^2 + 1)/x.
Step 2: Simplify the second term, (x^2+5x)/(x^2+4x-5).
We can factor the denominator to make it easier to work with. The denominator factors as (x-1)(x+5).
So, the expression becomes (x^2+5x)/((x-1)(x+5)).
Step 3: Combine the simplified terms from Step 1 and Step 2.
The expression is now ((x^2 + 1)/x) + ((x^2+5x)/((x-1)(x+5))).
To add these fractions, we need a common denominator. The common denominator is x((x-1)(x+5)).
To get the first term's denominator to match the common denominator, we multiply the numerator and denominator by (x-1)(x+5).
This gives us ((x^2 + 1)(x-1)(x+5))/(x((x-1)(x+5))).
For the second term, the denominator already matches the common denominator, so we don't need to make any changes.
The expression is now ((x^2 + 1)(x-1)(x+5))/(x((x-1)(x+5))) + ((x^2+5x)/((x-1)(x+5))).
Step 4: Combine the fractions by adding the numerators.
The numerator becomes ((x^2 + 1)(x-1)(x+5)) + (x^2+5x).
Simplifying the numerator gives us (x^3 - x^2 + 5x - 5) + (x^2 + 5x).
Combining like terms, we get x^3 + 5.
Therefore, the simplified expression is x^3 + 5.