$$\Large p(x) = ax^3 + bx^2 + cx =d$$

$$\Large p(0) = d$$

$$\Large p(1) = a + b + c + d$$

$$\Large p'(x) = 3ax^2 + 2bx + c$$

$$\Large p'(0) = c = \frac{p(1) - p(-1)}{2}$$

$$\Large p'(1) = 3a + 2b +c = \frac{p(2) - p(0)}{2}$$

$$\Large \left{ \begin{matrix} \begin{align*} a &= \frac{3p(0)+p(2)-3p(1)-p(-1))}{2}\\ b &= p(-1) + 2p(1) - \left(\frac{5p(0)+p(2)}{2}\right)\\ c &= \frac{1}{2}p(1)-\frac{1}{2}p[-1] \\ d &= p(0) \end{align*} \end{matrix} \right. $$