The characteristic polynomial is given by

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

In this example we have:

$$ \begin{aligned} p(\lambda) &= det(A - \lambda I) = det \left( \left[ \begin{matrix}-1&2&4&1\\5&3&1&1\\3&7&9&3\\2&-1&2&4\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 & 0 \\ 0 & \lambda & 0 & 0 \\ 0 & 0 & \lambda & 0 \\ 0 & 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}-1 - \lambda &2&4&1\\5&3 - \lambda &1&1\\3&7&9 - \lambda &3\\2&-1&2&4 - \lambda \end{matrix} \right| = \\ & = \end{aligned} $$

Characteristic polynomial calculator