Determine all the complex numbers $w = a + bi$ with $a, b \in \mathbb{R}$, such that there exists a polinomial $p(z)$ whose coefficients are real and positive such that $p(w) = 0.$
1. **Case 1: \( w \) is a non-negative real number**

   If \( w \) is a non-negative real number, then there is no polynomial \( p(z) \) with positive real coefficients such that \( p(w) = 0 \). This is because a polynomial with positive real coefficients cannot have a non-negative real root.

2. **Case 2: \( w \) is a negative real number**

   If \( w \) is a negative real number, then the polynomial \( p(z) = z - w \) works. For example, if \( w = -c \) where \( c > 0 \), then \( p(z) = z + c \) has positive real coefficients and \( p(-c) = 0 \).

3. **Case 3: \( w = re^{i\theta} \) with \( \theta \) not a multiple of \( \pi \)**

   Suppose \( w = re^{i\theta} \) with \( \theta \) not a multiple of \( \pi \). Since \( w \) is a root of \( p(z) \) (where \( p(z) \) has positive real coefficients) if and only if \( \bar{w} \) is, we may assume that \( 0 < \theta < \pi \).

4. **Finding the smallest \( m \) such that \( m\theta > \frac{\pi}{2} \)**

   Let \( m \) be the smallest natural number such that \( m\theta > \frac{\pi}{2} \). We then note that \( (m - 1)\theta \leq \frac{\pi}{2} \), and so:
   \[
   m\theta \leq \frac{\pi}{2} + \theta < \frac{\pi}{2} + \pi = \frac{3\pi}{2}
   \]
   We thus see that \( w^m \) has a negative real part.

5. **Constructing the polynomial \( q(z) \)**

   Note that \( w^m \) is a root of the polynomial:
   \[
   q(z) = z^2 - 2 \Re(w^m) z + |w|^{2m}
   \]
   which has positive real coefficients because \( \Re(w^m) \) is negative and \( |w|^{2m} \) is positive.

6. **Constructing the polynomial \( p(z) \)**

   The number \( w \) is then a root of:
   \[
   p(z) = q(z^m) = z^{2m} - 2 \Re(w^m) z^m + |w|^{2m}
   \]
   which has non-negative real coefficients. 

7. **Ensuring positive coefficients**

   To ensure that the polynomial has positive coefficients, we multiply by:
   \[
   (z^{m - 1} + z^{m - 2} + \cdots + z + 1)
   \]
   The resulting polynomial:
   \[
   (z^{m - 1} + z^{m - 2} + \cdots + z + 1) p(z)
   \]
   can be seen to have positive coefficients.

The final answer is \( \boxed{ w } \) can be any negative real number or any complex number \( w = re^{i\theta} \) with \( 0 < \theta < \pi \).