If you want to divide polynomials, you do it sorta like long division. Let’s take: for this purpose, | will be the symbol for long division, like the |‾‾‾‾ symbol in gradeschool.

 
     __x__+__6____
x-2 | x^2 + 4x  + 3    so x goes into x^2 once. now we multiply x by the entire left-side polynomial.
    -(x^2 - 2x)        x * x-2 = x^2-2x
      0   - 6x  + 3    6 b/c 4- -2 = 4+2;   carry down the 3.
          - (6x - 12)  6*x-2 = 6x - 12
            0   + 15   remainder of 15.
 
 
Answer:   x+6+(15/x-2)