$\begin{align*}
p(x) &= a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x^1 + a_0 \\
 &= (a_n x + a_{n-1}) x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x^1 + a_0 \\
 &= ((a_n x + a_{n-1}) x + a_{n-2}) x^{n-2} + \dots + a_1 x^1 + a_0 \\
 &\ \vdots \\
 &= (\dots((a_n x + a_{n-1}) x + a_{n-2}) x + \dots + a_1) x + a_0
\end{align*}$