Real roots and derivative polynomial

For the first part, use Rolle’s Theorem and the fact that if a is a root of P with the order p then it is a root of P' of order p-1 so if we have the roots $a_{1}$,...,$a_{m}$ with multiplicities
$n_{1}$,...,$n_{m}$, we have m-1 roots for P' that are not those of P with Rolle and $n_{i}$-1 from each root $a_{i}$ of P so in total
$$\sum_{i=1}^{m} (n_i - 1) + m - 1 = \left( \sum_{i=1}^{m} n_i \right) - 1 = \deg(P) - 1 = \deg(P')$$

The second point is wrong, just consider $X^{2}$+1 for P.