$$\Large p(x) = ax^3 + bx^2 + cx =d$$
$$\Large p(0) = d$$
$$\Large p(1) = a + b + c + d$$
$$\Large p'(x) = 3ax^2 + 2bx + c$$
$$\Large p'(0) = c = \frac{p(1) - p(-1)}{2}$$
$$\Large p'(1) = 3a + 2b +c = \frac{p(2) - p(0)}{2}$$
$$\Large
\left {
\begin{matrix}
\begin{align*}
a &= \frac{3p(0)+p(2)-3p(1)-p(-1)}{2}\\
b &= p(-1) + 2p(1) - \left(\frac{5p(0)+p(2)}{2}\right)\\
c &= \frac{1}{2}p(1)-\frac{1}{2}p(-1) \\
d &= p(0)
\end{align*}
\end{matrix}
\right.$$
