where did come from?? has no meaning because the RHS of the inequality is not a number!!

the inverse of a polynomial is not necessarily a polynomial. so and might not even exist in your polynomial ring!

let d(x) be their GCD, then we have:

f(x)=a(x).d(x)

g(x)=b(x).d(x)

so where and are polynomials u(x),v(x)

the problem, as you posted, doesn't make any sense and you certainly won't get help if your problem has itself a problem!