Proof

If the residue class a + (d) K[X]/(d) has inverse b + (d), then ab = 1 (mod d). Hence there is a polynomial p with

ab + pd = 1.

But that implies that gcd(a,d) = 1.

On the other hand, if gcd(a,d) = 1, then the extended Euclidean algorithm leads to a method for finding polynomials b and p such that ab + pd = 1. But then b represents an inverse of the residue class a + (d).

The second statement is a trivial consequence of the first.