How is the process for simplifying 2x(x+1)/4(x+1) similar? Different?
The process for simplifying rational expressions really is the same as the process as simplifying fractions.
For example, if you were going to simplify the fraction 5/20, you would find a common divisor and divide that piece out of both the numerator and the denominator. In this example, 5 would be a common divisor.
5/20 = 1(5) / 4(5) = 1/4
In the example you listed above, there is a term that shows up in both the numerator and the denominator. To simplify the expression, you need to divide that out of both pieces. I also see an even integer in both the numerator and denominator, so you know right away that they have a common divisor that can be taken out.