How to do equations with variables with the same coefficient on each side?

Example 1:
4x + 2 = 4x + 4subtract 4x from both sides
4x - 4x + 2 = 4x - 4x + 4...simplify
2 = 4 (incorrect)

example 2 :
4x + 2 = 4x + 2...subtract 4x from both sides
4x - 4x + 2 = 4x - 4x + 2...simplify
2 = 2 (correct)

** if the variables cancel outand u r left with an incorrect statement (like in example 1), then ur equations will have no solution
** if the variables cancel out...and u r left with a correct statement (like in example 2), then ur equations will have infinite solutions