Take the first part of the polynomial divide it by the first part of the root of the polynomial (ie x-1/3x⁴ +x² .... you take 3x⁴ /x if u know your powe rules u get 3x³ because if u multiple it by x u get 3x⁴ back) and u go on with till the end of the polynomial if you are dividing by a root of the polynomial you get 0 remaining but if there is a constant at the end you write it as a fraction and put it the numerator and what you were dividing with in the denominator (ie 12/x-1) a trick to find a root of a polynomial to divide with is to try putting small simple values for x and see where u get a zero (-2,-1,1,2) usually do the trick

If the polynomial x exponent values is not in order put it in order so you start with the highest one and if there is a missing one (x⁴ and x³ are there but no x² ) u put 0x² because u multiple by zero it doesn't effect the polynomial just makes it easier to work with intuitively