Sure, let's break it down into clear steps.
Step 1: Rationalize the Expressions
First, we need to simplify each rational expression. Rational expressions, just like fractions, can often be reduced to their simplest form, which makes the following steps easier.
For the first expression, (x² - 25)/(x - 5), simplifying it involves taking advantage of the fact that the numerator is a difference of squares. Thus, we rewrite (x² - 25) as (x - 5)(x + 5), so the first expression becomes:
(x - 5)(x + 5) / (x - 5)
The (x - 5) terms cancel out, leaving us with (x + 5).
For the second expression, (x + 5)/(4x - 20), we can simplify by factoring out a 4 from the denominator, getting:
(x + 5) / 4*(x - 5)
Step 2: Perform the Division
Now, we have to divide the first simplified expression by the second. It's important to remember here that dividing by a fraction is the same as multiplying by its reciprocal. So, we'll change the division to multiplication and invert the second fraction:
(x + 5) * (4*(x - 5))/(x + 5)
The (x + 5) terms cancel out, leaving us with 4*(x - 5), which simplifies to:
4*x - 20
So, the result of the original division problem is 4*x - 20.
So, the simplified expressions and the result of the division are (x + 5), (x + 5)/(4*(x - 5)), and 4*x - 20 respectively.