The characteristic polynomial is given by

$$ p(\lambda) = det(A - \lambda I) $$

In this example we have:

$$ \begin{aligned} p(\lambda) &= det(A - \lambda I) = det \left( \left[ \begin{matrix}1&1&0&0\\0&1&0&0\\1&2&3&1\\-2&-5&-4&-1\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}1 - \lambda &1&0&0\\0&1 - \lambda &0&0\\1&2&3 - \lambda &1\\-2&-5&-4&-1 - \lambda \end{matrix} \right| = \\ & = \end{aligned} $$

Characteristic polynomial calculator