Factor or multiply out terms and expressions to create as many like terms as possible. If your terms aren't unlike, it's impossible to combine them with any of the four basic operations. For example, for the equation [4xy^2 (x^2 + 3x)] / [2xy (x+3)], factor the x out of the (x^2 + 3x) to make the numerator 4xy^2 (x+3).
Divide out any like terms, starting with constants. The numerator has a 4 and the denominator has a 2. Because 4 = 2^2, you can cross out the 2 in the denominator and change the 4 in the numerator to a 2.
Cancel out any possible exponential variables. Because there is an "x" in both numerator and denominator, you can cross both of them out. Also, mark through the "x+3" in both numerator and denominator. The "y^2" in the numerator would become a "y" and the "y" in the denominator would go away. The end result would be 2y -- much simpler than what we started with.