It's pretty straight forward:
suppose r is the double root of the given polynomial, thus:
now according to the theorem you've already stated:
If a polynomial has a root \alpha of multiplicity m, then P'(x) has the root \alpha with multiplicity (m-1)
so r must be one of the roots of P'(x), thus:
r=0 is a double root of P(x) only if d = 0, which is not necessarily true, so we'll choose:
now substitute this value of r in the first equation:
... simplify and obtain the answer.